Counting Practice

1.Multiply the choices

When one choice is followed by another, and each is free of the other, multiply the number of options. This is the counting principle, and everything else on this sheet is built on it.

How many?

1. 1.A lunch counter offers 3 sandwiches and 4 drinks. How many different sandwich-and-drink lunches can be ordered?
2. 2.There are 5 buses into town in the morning and 4 buses home in the evening. In how many ways can you choose one bus in and one bus back?
3. 3.You have 4 shirts, 3 pairs of trousers and 2 pairs of shoes. How many different outfits of one of each can you make?
4. 4.A set dinner is one starter, one main course and one dessert. There are 3 starters, 5 mains and 2 desserts. How many different dinners are there?
5. 5.There are 7 buses into town in the morning and 6 buses home in the evening. In how many ways can you choose one bus in and one bus back?
6. 6.You have 5 shirts, 4 pairs of trousers and 3 pairs of shoes. How many different outfits of one of each can you make?
7. 7.A set dinner is one starter, one main course and one dessert. There are 4 starters, 6 mains and 3 desserts. How many different dinners are there?
8. 8.A lock has 3 dials, and each dial can be set to any digit from 0 to 9. How many different settings are there?

2.Putting things in a row

Every thing goes in, and the question is only the order. The first place can be filled by any of them, the next by any of those left, and so on down to 1. That product is written with an exclamation mark and called a factorial.

How many orders?

1. 9.Work out 3!  (that is, 3 × 2 × 1).
2. 10.Work out 4!  (that is, 4 × 3 × 2 × 1).
3. 11.In how many different orders can 4 people stand in a line?
4. 12.5 people are each going to speak once at a meeting. In how many different orders can they speak?
5. 13.In how many different orders can 6 different books be put on a shelf?
6. 14.In how many different orders can 7 people stand in a line?

3.When the order matters

Only some of them are chosen, and swapping two of the chosen ones makes a different outcome — a different medal, a different job. Multiply down, one fewer each time, for as many places as there are to fill. These are permutations.

How many ways?

1. 15.A group of 5 people needs one person to chair a meeting and a different person to take notes. In how many ways can the two jobs be filled?
2. 16.6 people have asked to speak at a meeting. There is time for only two of them, and someone has to decide who goes first and who goes second. In how many ways can that be decided?
3. 17.5 runners are in a race. In how many different ways can the gold, silver and bronze medals be given out?
4. 18.A door code is 2 digits long, uses the digits 0 to 9, and never uses the same digit twice. How many different codes are possible?
5. 19.A union local of 6 stewards chooses a president, a vice-president and a secretary from among them. In how many ways can this be done?
6. 20.8 runners are in a race. In how many different ways can the gold, silver and bronze medals be given out?
7. 21.9 people have asked to speak at a meeting. There is time for only three of them, and someone has to decide who goes first, who goes second and who goes third. In how many ways can that be decided?
8. 22.A door code is 3 digits long, uses the digits 0 to 9, and never uses the same digit twice. How many different codes are possible?

4.When the order does not matter

Only some are chosen, and swapping two of the chosen ones changes nothing — the same people are on the committee. Multiply down as before, then divide by the number of ways the chosen ones could be put in order. These are combinations.

How many ways?

1. 23.A manager has to pick 2 of 5 workers to cover a Saturday. How many different pairs could she pick?
2. 24.You own 6 books and have room to take 2 on a trip. How many different pairs could you take?
3. 25.In how many ways can a committee of 3 be chosen from 6 people?
4. 26.A tenants’ association of 7 members sends two of them to a meeting with the landlord. How many different groups could it send?
5. 27.A pizza place has 8 toppings, and you are allowed two. How many different choices of toppings are there?
6. 28.In how many ways can a committee of 3 be chosen from 8 people?
7. 29.A pizza place has 9 toppings, and you are allowed three. How many different choices of toppings are there?
8. 30.A tenants’ association of 10 members sends three of them to a meeting with the landlord. How many different groups could it send?

5.You decide

These are mixed. Before any arithmetic, swap two of the chosen things in your head and ask whether that makes a different outcome. Write down order matters or order does not matter first; then work it out. The key shows which it is.

Say whether order matters, then say how many.

1. 31.In how many ways can a committee of 3 be chosen from 7 people?
2. 32.A group of 8 people needs one person to chair a meeting and a different person to take notes. In how many ways can the two jobs be filled?
3. 33.A tenants’ association of 6 members sends three of them to a meeting with the landlord. How many different groups could it send?
4. 34.A union local of 7 stewards chooses a president, a vice-president and a secretary from among them. In how many ways can this be done?
5. 35.You own 7 books and have room to take 2 on a trip. How many different pairs could you take?
6. 36.6 runners are in a race. In how many different ways can the gold, silver and bronze medals be given out?
7. 37.A pizza place has 10 toppings, and you are allowed two. How many different choices of toppings are there?
8. 38.7 people have asked to speak at a meeting. There is time for only two of them, and someone has to decide who goes first and who goes second. In how many ways can that be decided?
9. 39.A manager has to pick 2 of 8 workers to cover a Saturday. How many different pairs could she pick?
10. 40.A coach has 9 players and has to decide who bats first and who bats second. In how many ways can she decide?