Year 2: Arranging objects and making combinations
Part of the Year 2 position and direction chapter. Chapter overview · whole chapter on one page
Arranging objects and making combinations
We can arrange objects in different orders. Each different order is a different arrangement.
We can also make combinations. A combination picks one thing from each group.
To find every one, make a list. Then count each one once.
Key ideas
- Each different order of the same objects is an arrangement.
- Make a list of every order, so that none is missed.
- A combination picks one thing from each group, such as one hat and one coat.
- Count each pair once. Do not count it twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.star, ball; ball, star
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 3 coats.hat 1 with coat 1; hat 1 with coat 2; hat 1 with coat 3
- Take each of the other hats. Pair each one with every coat too.hat 2 with coat 1; hat 2 with coat 2; hat 2 with coat 3
- Count every outfit once. There are 2 hats and 3 coats.6 outfits
- Write the answer. 6
Check. Count the outfits again, hat by hat: 6. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube, apple; ball, apple, cube; cube, ball, apple; cube, apple, ball; apple, ball, cube; apple, cube, ball
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Watch out
Watch out: a ball then a cube is a different arrangement from a cube then a ball. The order matters.
Exam tip
Grown-up tip. Use three toys. Ask: “How many ways can we line them up?” Write each line on paper, then count the lines.
Exercise 9B Arranging objects and combinations
Non-calculatorMethod. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.cup, bag; bag, cup
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 2 coats.hat 1 with coat 1; hat 1 with coat 2
- Take each of the other hats. Pair each one with every coat too.there are no other hats
- Count every outfit once. There are 1 hats and 2 coats.2 outfits
- Write the answer. 2
Check. Count the outfits again, hat by hat: 2. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Pick two objects at a time. Write each pair once, then count the pairs.
- List the pairs. Each pair is written only once.ball and cube; ball and apple; cube and apple
- Count the pairs. Count each pair once.3 pairs
- Write the answer. 3
Check. ball and cube is the same pair as cube and ball, so it is counted once. ✓
Watch out. The order of the two does not matter when you only pick them.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 2 coats.hat 1 with coat 1; hat 1 with coat 2
- Take each of the other hats. Pair each one with every coat too.hat 2 with coat 1; hat 2 with coat 2; hat 3 with coat 1; hat 3 with coat 2; hat 4 with coat 1; hat 4 with coat 2
- Count every outfit once. There are 4 hats and 2 coats.8 outfits
- Write the answer. 8
Check. Count the outfits again, hat by hat: 8. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.car, bus, bike; car, bike, bus; bus, car, bike; bus, bike, car; bike, car, bus; bike, bus, car
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube; cube, ball
- Count the orders. There are 2 different orders.2 arrangements
- Write the answer. 2
Check. Each object can go at the front in 1 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Put each object at the front in turn, then write every order of the rest.
- List every order. Each order is one arrangement.ball, cube, apple; ball, apple, cube; cube, ball, apple; cube, apple, ball; apple, ball, cube; apple, cube, ball
- Count the orders. There are 6 different orders.6 arrangements
- Write the answer. 6
Check. Each object can go at the front in 2 orders, so the list is complete. ✓
Watch out. Two arrangements are different if the objects are in a different order, even though they are the same objects.
Method. Take each hat in turn. Pair it with every coat, then count the pairs.
- Take the first hat. It can be worn with each of the 3 coats.hat 1 with coat 1; hat 1 with coat 2; hat 1 with coat 3
- Take each of the other hats. Pair each one with every coat too.there are no other hats
- Count every outfit once. There are 1 hats and 3 coats.3 outfits
- Write the answer. 3
Check. Count the outfits again, hat by hat: 3. ✓
Watch out. Make sure every hat is paired with every coat, and no pair is counted twice.
Method. Count the balls in the first 9 objects, one at a time.
- Write the first objects in order. Count from the first object.1 star, 2 ball, 3 ball, 4 star, 5 ball, 6 ball, 7 star, 8 ball, 9 ball
- Find the balls. The balls are in places 2, 3, 5, 6, 8, 9.places 2, 3, 5, 6, 8, 9
- Count them. There are 6 balls.6
Check. The repeat is star, ball, ball, so each part has 2 ball(s). ✓
Watch out. Count the places you listed. Do not count the other objects.
Method. Count the ways to arrange a star and a ball, then compare the number with the claim.
- Count the ways to arrange a star and a ball. There are 2 of them.2
- Compare with the claim. The claim says 4.2 is not the same as 4
- Decide. Is the claim right?No
Check. Count them again to check: 2. ✓
Watch out. Count every one, then compare. Do not guess from the picture.
Method. Count the ways to arrange a bag and a hat, then compare the number with the claim.
- Count the ways to arrange a bag and a hat. There are 2 of them.2
- Compare with the claim. The claim says 1.2 is not the same as 1
- Decide. Is the claim right?No
Check. Count them again to check: 2. ✓
Watch out. Count every one, then compare. Do not guess from the picture.