GCSE Maths: Systematic listing and the product rule (Higher)
Part of the GCSE Maths factors, multiples and primes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Systematic listing and the product rule
Many questions ask “how many different ways ...?” How many outfits can you make? How many different two-digit numbers? How many ways can you pay 30p? To answer, you need to find all the possibilities without missing any and without counting any twice.
The trick is to be systematic. A systematic list uses a plan:
- Decide the order of the choices (for example, T-shirt first, then trousers).
- Fix the first choice and list every way to make the other choices.
- Move to the next first choice and repeat.
- Count at the end.
For numbers, list them in order of size: start with the smallest first digit, then the smallest second digit, and so on.
Useful ideas: for totals (such as two dice adding to 7) go through the first number in order and work out what the second must be. For coin questions start with as many of the biggest coin as you can, then use one fewer each time.
Counting without listing. Look at the tree: 3 colours, each with 2 choices, gives 3 × 2 = 6. This is a shortcut: if there are m ways to do one thing and n ways to do another, you can multiply to count the total.
The product rule for counting says: if there are m ways of doing one task and, for each of these, n ways of doing another task, then the total number of ways of doing both is m × n. It extends to more tasks: multiply the number of choices at every stage. For example, a 4-digit PIN (digits 0 to 9) has 10 choices for each digit: 10 × 10 × 10 × 10 = 10 000 PINs. If digits cannot be repeated, the choices go down each time: 10 × 9 × 8 × 7 = 5040. When one position has a restriction (such as “even”), fill that position first.
Key ideas
- Be systematic: choose an order, fix the first choice, list the rest, then move on.
- List numbers in order of size so nothing is missed.
- Count only when the list is finished.
- A quick check: with m choices then n choices there are m × n combinations.
- Product rule (Higher): multiply the number of choices at each stage.
- Fill any restricted position first.
Key words. A combination here is one choice from each group. A systematic list is a list written in a planned order.
Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.
- Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of colour first.3 colour, 2 bottoms
- Take the first choice of colour and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.Red: Jeans, Shorts
- Repeat for each of the other choices of colour. Work through the remaining choices in order.Blue: Jeans, Shorts
Green: Jeans, Shorts - Count the combinations. Count every item in your list.Total = 6
Check. Multiply the number of choices: 3 × 2 = 6. This is the product rule and it agrees with the list. ✓
Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used only once.Digits: 2, 5, 7 Length: 3 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 2: 257, 275
Starting with 5: 527, 572
Starting with 7: 725, 752 - Count them. Count every number in the list.Total = 6
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.
Method. Make a systematic list: take the first number in order and find the second number that gives 8.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: no second number works
First = 2: second = 6 (2 + 6 = 8) ✓
First = 3: second = 5 (3 + 5 = 8) ✓
First = 4: second = 4 (4 + 4 = 8) ✓
First = 5: second = 3 (5 + 3 = 8) ✓
First = 6: second = 2 (6 + 2 = 8) ✓ - Write the pairs. List them as (first, second).(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
- Count them. Count the pairs.5 pairs
Check. Each pair really gives 8: 2 + 6 = 8, 3 + 5 = 8, 4 + 4 = 8, 5 + 3 = 8, 6 + 2 = 8. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Make a systematic list for making 30p. Fix the number of the biggest coin, then list the ways to make the rest.
- Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.Coins: 20p, 10p, 5p
- Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.1 × 20p + 1 × 10p = 30p
1 × 20p + 2 × 5p = 30p
3 × 10p = 30p
2 × 10p + 2 × 5p = 30p
1 × 10p + 4 × 5p = 30p
6 × 5p = 30p - Count the ways. Count the lines.6 ways
Check. Each line adds up to 30p, and no two lines are the same. ✓
Watch out. Be organised: write the number of each coin in the same order every time.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st digit: 10 choices (any digit)
2nd digit: 9 choices (not the same as the 1st)
3rd digit: 8 choices (not either of the first two)
4th digit: 7 choices (not any of the first three) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.10 × 9 × 8 × 7 = 5040
- Write the answer. State the total number of ways.There are 5040 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Units digit (fill this first): 2 choices (must be 2 or 4)
Hundreds digit: 4 choices (5 digits, but one is used in the units)
Tens digit: 3 choices (two digits are already used) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.2 × 4 × 3 = 24
- Write the answer. State the total number of ways.There are 24 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Watch out
Do not just write down numbers at random. A messy list almost always misses some. Also check the rules: can digits be repeated? Does the order matter (12 and 21 are different numbers)? And with the product rule, make sure you multiply the numbers of choices. Do not add them.Exercise 2E Systematic listing and the product rule
Non-calculatorKey words. A combination here is one choice from each group. A systematic list is a list written in a planned order.
Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.
- Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of starters first.3 starters, 2 mains
- Take the first choice of starters and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.soup: fish, pasta
- Repeat for each of the other choices of starters. Work through the remaining choices in order.salad: fish, pasta
pate: fish, pasta - Count the combinations. Count every item in your list.Total = 6
Check. Multiply the number of choices: 3 × 2 = 6. This is the product rule and it agrees with the list. ✓
Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.
Key words. A combination here is one choice from each group. A systematic list is a list written in a planned order.
Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.
- Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of shirts first.2 shirts, 3 ties
- Take the first choice of shirts and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.white: red, green, grey
- Repeat for each of the other choices of shirts. Work through the remaining choices in order.blue: red, green, grey
- Count the combinations. Count every item in your list.Total = 6
Check. Multiply the number of choices: 2 × 3 = 6. This is the product rule and it agrees with the list. ✓
Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.
Key words. A combination here is one choice from each group. A systematic list is a list written in a planned order.
Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.
- Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of colours first.4 colours, 2 engines
- Take the first choice of colours and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.red: 1.0, 1.4
- Repeat for each of the other choices of colours. Work through the remaining choices in order.black: 1.0, 1.4
silver: 1.0, 1.4
white: 1.0, 1.4 - Count the combinations. Count every item in your list.Total = 8
Check. Multiply the number of choices: 4 × 2 = 8. This is the product rule and it agrees with the list. ✓
Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used only once.Digits: 1, 2, 3 Length: 3 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 1: 123, 132
Starting with 2: 213, 231
Starting with 3: 312, 321 - Count them. Count every number in the list.Total = 6
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used more than once.Digits: 4, 6, 9 Length: 2 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 4: 44, 46, 49
Starting with 6: 64, 66, 69
Starting with 9: 94, 96, 99 - Count them. Count every number in the list.Total = 9
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used only once.Digits: 1, 3, 5, 7 Length: 2 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 1: 13, 15, 17
Starting with 3: 31, 35, 37
Starting with 5: 51, 53, 57
Starting with 7: 71, 73, 75 - Count them. Count every number in the list.Total = 12
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used only once. The number must be even, so the last digit must be even.Digits: 1, 2, 3, 4 Length: 3 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 1: 124, 132, 134, 142
Starting with 2: 214, 234
Starting with 3: 312, 314, 324, 342
Starting with 4: 412, 432 - Count them. Count every number in the list.Total = 12
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.
Method. Make a systematic list: take the first number in order and find the second number that gives 7.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: second = 6 (1 + 6 = 7) ✓
First = 2: second = 5 (2 + 5 = 7) ✓
First = 3: second = 4 (3 + 4 = 7) ✓
First = 4: second = 3 (4 + 3 = 7) ✓
First = 5: second = 2 (5 + 2 = 7) ✓
First = 6: second = 1 (6 + 1 = 7) ✓ - Write the pairs. List them as (first, second).(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
- Count them. Count the pairs.6 pairs
Check. Each pair really gives 7: 1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7, 4 + 3 = 7, 5 + 2 = 7, 6 + 1 = 7. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Make a systematic list: take the first number in order and find the second number that gives 5.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: second = 4 (1 + 4 = 5) ✓
First = 2: second = 3 (2 + 3 = 5) ✓
First = 3: second = 2 (3 + 2 = 5) ✓
First = 4: second = 1 (4 + 1 = 5) ✓
First = 5: no second number works
First = 6: no second number works - Write the pairs. List them as (first, second).(1, 4), (2, 3), (3, 2), (4, 1)
- Count them. Count the pairs.4 pairs
Check. Each pair really gives 5: 1 + 4 = 5, 2 + 3 = 5, 3 + 2 = 5, 4 + 1 = 5. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Make a systematic list: take the first number in order and find the second number that gives 10.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: no second number works
First = 2: no second number works
First = 3: no second number works
First = 4: second = 6 (4 + 6 = 10) ✓
First = 5: second = 5 (5 + 5 = 10) ✓
First = 6: second = 4 (6 + 4 = 10) ✓ - Write the pairs. List them as (first, second).(4, 6), (5, 5), (6, 4)
- Count them. Count the pairs.3 pairs
Check. Each pair really gives 10: 4 + 6 = 10, 5 + 5 = 10, 6 + 4 = 10. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Make a systematic list: take the first number in order and find the second number that gives 12.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: no second number works
First = 2: second = 6 (2 × 6 = 12) ✓
First = 3: second = 4 (3 × 4 = 12) ✓
First = 4: second = 3 (4 × 3 = 12) ✓
First = 5: no second number works
First = 6: second = 2 (6 × 2 = 12) ✓ - Write the pairs. List them as (first, second).(2, 6), (3, 4), (4, 3), (6, 2)
- Count them. Count the pairs.4 pairs
Check. Each pair really gives 12: 2 × 6 = 12, 3 × 4 = 12, 4 × 3 = 12, 6 × 2 = 12. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Make a systematic list for making 20p. Fix the number of the biggest coin, then list the ways to make the rest.
- Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.Coins: 10p, 5p
- Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.2 × 10p = 20p
1 × 10p + 2 × 5p = 20p
4 × 5p = 20p - Count the ways. Count the lines.3 ways
Check. Each line adds up to 20p, and no two lines are the same. ✓
Watch out. Be organised: write the number of each coin in the same order every time.
Method. Make a systematic list for making 40p. Fix the number of the biggest coin, then list the ways to make the rest.
- Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.Coins: 20p, 10p
- Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.2 × 20p = 40p
1 × 20p + 2 × 10p = 40p
4 × 10p = 40p - Count the ways. Count the lines.3 ways
Check. Each line adds up to 40p, and no two lines are the same. ✓
Watch out. Be organised: write the number of each coin in the same order every time.
Method. Make a systematic list for making 30p. Fix the number of the biggest coin, then list the ways to make the rest.
- Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.Coins: 20p, 10p, 5p
- Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.1 × 20p + 1 × 10p = 30p
1 × 20p + 2 × 5p = 30p
3 × 10p = 30p
2 × 10p + 2 × 5p = 30p
1 × 10p + 4 × 5p = 30p
6 × 5p = 30p - Count the ways. Count the lines.6 ways
Check. Each line adds up to 30p, and no two lines are the same. ✓
Watch out. Be organised: write the number of each coin in the same order every time.
Method. Fix the first position, list all the ways to order the rest, then move to the next person.
- Decide on a system. Fix who is first. Then list every order for the rest, one person at a time. This way no order can be missed or repeated.People: Amy, Ben, Cal
- List the orders for each choice of first person. Work through the people in order.Amy first: Amy Ben Cal, Amy Cal Ben
Ben first: Ben Amy Cal, Ben Cal Amy
Cal first: Cal Amy Ben, Cal Ben Amy - Count the orders. Total = 6
Check. 3 choices for first, then 2, then ...: 3 × 2 × 1 = 6. ✓
Watch out. A different order is a different arrangement: ABC and ACB are not the same.
Key words. A combination here is one choice from each group. A systematic list is a list written in a planned order.
Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.
- Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of starters first.2 starters, 3 mains, 2 desserts
- Take the first choice of starters and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.S1: M1 + D1, M1 + D2, M2 + D1, M2 + D2, M3 + D1, M3 + D2
- Repeat for each of the other choices of starters. Work through the remaining choices in order.S2: M1 + D1, M1 + D2, M2 + D1, M2 + D2, M3 + D1, M3 + D2
- Count the combinations. Count every item in your list.Total = 12
Check. Multiply the number of choices: 2 × 3 × 2 = 12. This is the product rule and it agrees with the list. ✓
Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.
Method. Make a systematic list for making 60p. Fix the number of the biggest coin, then list the ways to make the rest.
- Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.Coins: 50p, 20p, 10p
- Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.1 × 50p + 1 × 10p = 60p
3 × 20p = 60p
2 × 20p + 2 × 10p = 60p
1 × 20p + 4 × 10p = 60p
6 × 10p = 60p - Count the ways. Count the lines.5 ways
Check. Each line adds up to 60p, and no two lines are the same. ✓
Watch out. Be organised: write the number of each coin in the same order every time.
Method. Make a systematic list: take the first number in order and find the second number that gives 5.
- Go through the first number in order. For each first number, work out what the second number would have to be.First = 1: second = 4 (1 + 4 = 5) ✓
First = 2: second = 3 (2 + 3 = 5) ✓
First = 3: second = 2 (3 + 2 = 5) ✓ - Write the pairs. List them as (first, second).(1, 4), (2, 3), (3, 2)
- Count them. Count the pairs.3 pairs
Check. Each pair really gives 5: 1 + 4 = 5, 2 + 3 = 5, 3 + 2 = 5. ✓
Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.
Method. Count the outfits by listing, so you can see the mistake.
- List the outfits for T-shirt 1. T-shirt 1 can be worn with either pair of trousers.T1 + trousers A, T1 + trousers B (2 outfits)
- Do the same for T-shirts 2 and 3. Each T-shirt has the same 2 choices.T2 + A, T2 + B, T3 + A, T3 + B
- Count. Each of the 3 T-shirts gives 2 outfits.3 × 2 = 6 outfits, not 5
Check. A tree diagram has 3 branches, each splitting into 2, giving 6 ends. ✓
Watch out. Choices that happen one after another are multiplied. Adding is for “either this or that”.
Method. List the numbers in order, one first digit at a time, and compare with Ella’s list.
- Starting with 1. Then the other two digits in order.123, 132
- Starting with 2. Then the other two digits in order.213, 231
- Starting with 3. Then the other two digits in order.312, 321
- Compare. There should be 3 × 2 = 6 numbers. Ella has only 5.Missing: 231. Listing by first digit would have shown two numbers for each first digit.
Check. 6 numbers in total: 123, 132, 213, 231, 312, 321. ✓
Watch out. Always count how many you expect before you stop listing.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st digit: 10 choices
2nd digit: 10 choices
3rd digit: 10 choices
4th digit: 10 choices - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.10 × 10 × 10 × 10 = 10,000
- Write the answer. State the total number of ways.There are 10,000 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st digit: 10 choices
2nd digit: 9 choices (one digit used)
3rd digit: 8 choices (two digits used)
4th digit: 7 choices (three digits used) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.10 × 9 × 8 × 7 = 5040
- Write the answer. State the total number of ways.There are 5040 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Last digit (fill first): 5 choices (0, 2, 4, 6 or 8)
1st digit: 10 choices
2nd digit: 10 choices
3rd digit: 10 choices - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.5 × 10 × 10 × 10 = 5000
- Write the answer. State the total number of ways.There are 5000 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st digit: 9 choices (any digit except 0)
2nd digit: 10 choices
3rd digit: 10 choices
4th digit: 10 choices - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.9 × 10 × 10 × 10 = 9000
- Write the answer. State the total number of ways.There are 9000 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Last digit (fill first): 5 choices (1, 3, 5, 7 or 9)
1st digit: 9 choices (any digit except 0)
2nd digit: 10 choices - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.5 × 9 × 10 = 450
- Write the answer. State the total number of ways.There are 450 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st letter: 26 choices
2nd letter: 25 choices (not the first)
3rd letter: 24 choices (not the first two) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.26 × 25 × 24 = 15,600
- Write the answer. State the total number of ways.There are 15,600 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Units digit (fill first): 3 choices (2, 4 or 6)
Hundreds digit: 5 choices (six digits, one already used)
Tens digit: 4 choices (two digits already used) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.3 × 5 × 4 = 60
- Write the answer. State the total number of ways.There are 60 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Hundreds digit (fill first): 3 choices (4, 5 or 6)
Tens digit: 5 choices (one digit used)
Units digit: 4 choices (two digits used) - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.3 × 5 × 4 = 60
- Write the answer. State the total number of ways.There are 60 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.Hundreds digit (fill first): 3 choices (1, 3 or 5)
Tens digit: 5 choices
Units digit: 4 choices - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.3 × 5 × 4 = 60
- Write the answer. State the total number of ways.There are 60 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says: if there are m ways to do one task and, for each, n ways to do another, there are m × n ways to do both.
Method. Use the product rule for counting: multiply the number of choices at each stage.
- Split the job into stages. The product rule works when the whole job is done one stage after another.1st place: 5 choices
2nd place: 4 choices
3rd place: 3 choices
4th place: 2 choices
5th place: 1 choice - Multiply the numbers of choices. If there are m ways to do the first stage and, for each of those, n ways to do the second, there are m × n ways altogether. The same idea extends to more stages.5 × 4 × 3 × 2 × 1 = 120
- Write the answer. State the total number of ways.There are 120 ways.
Check. A small case shows why: with 2 choices and then 3 choices there are 2 × 3 = 6 pairs, which you can list. ✓
Watch out. Fill in the stage with a restriction first (for example, the last digit of an even number), then the other stages.
Key words. The product rule for counting says multiply the number of choices at each stage.
Method. Place the restricted books first, then fill the middle.
- Choose which book goes at the left end. It must be A or B.2 choices
- The right end. It must be the other one of A and B.1 choice
- Fill the 3 middle places with the other 3 books. Any order: 3, then 2, then 1 choices.3 × 2 × 1 = 6
- Multiply the stages. Use the product rule.2 × 1 × 6 = 12
Check. There are 120 arrangements in all and 12 of these have A and B at the ends. ✓
Method. Use the product rule: multiply the number of choices for each of the five characters.
- Count the choices for each character. Letters can repeat, digits can repeat.Letter 1: 26, Letter 2: 26, Digit 1: 10, Digit 2: 10, Digit 3: 10
- Multiply. 26 × 26 = 676, and 676 × 10 × 10 × 10 = 676 000
- Compare with 600 000. The question says “show that”, so state the conclusion.676 000 > 600 000, so there are more than 600 000 passwords.
Check. 262 × 103 = 676 × 1000 = 676 000. ✓
Method. Write the numbers in order, one first digit at a time.
- Read the rules. Each digit can be used only once. The number must be bigger than 3000, so it must start with 3 or 4.Digits: 1, 2, 3, 4 Length: 4 digits
- List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.Starting with 3: 3124, 3142, 3214, 3241, 3412, 3421
Starting with 4: 4123, 4132, 4213, 4231, 4312, 4321 - Count them. Count every number in the list.Total = 12
Check. The list is in increasing order with no repeats, so every number is counted once. ✓
Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.