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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 8

Permutations

Use a permutation when the order of the choices makes a different outcome.

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Key Ideas Easy Medium Hard

nPr is a permutation: order matters.

nPr = n × (n − 1) × … for r factors

What to Remember

Order matters

A permutation is a ranking, a password, or a line of officers.

Captain then secretary is different from secretary then captain.

No repeats

If nothing can be used twice, multiply downward.

nPr = n × (n − 1) × … for r factors.

Level 1

Easy

Officers and rankings

Example
A club of 7 members chooses a president, a secretary, and a treasurer. How many ways can the offices be filled?

nPr = n × (n − 1) × … for r factors

7P3 = 7 × 6 × 5

210

Answer: 210

1
A team of 6 chooses a captain and a co-captain. How many ways can the two offices be filled?
Show Answer

nPr = n × (n − 1) × … for r factors

6P2 = 6 × 5

30

Answer: 30

2
Nine paintings will be hung in first place and second place. How many ways can the two places be filled?
Show Answer

nPr = n × (n − 1) × … for r factors

9P2 = 9 × 8

72

Answer: 72

3
Five actors are assigned the roles of first, second, and third speaker. How many ways can the roles be filled?
Show Answer

nPr = n × (n − 1) × … for r factors

5P3 = 5 × 4 × 3

60

Answer: 60

4
Eight runners finish in gold and silver only. How many ways can those two places be filled?
Show Answer

nPr = n × (n − 1) × … for r factors

8P2 = 8 × 7

56

Answer: 56

5
Ten students are ranked first, second, and third in a contest. How many ways can the three places be filled?
Show Answer

nPr = n × (n − 1) × … for r factors

10P3 = 10 × 9 × 8

720

Answer: 720

Level 2

Medium

A code with no repeats

Example
A lock uses 4 different numbers chosen from 1 through 12. How many codes are possible?

nPr = n × (n − 1) × … for r factors

12P4 = 12 × 11 × 10 × 9

11,880

Answer: 11,880

1
A code uses 3 different numbers chosen from 1 through 9. How many codes are possible?
Show Answer

nPr = n × (n − 1) × … for r factors

9P3 = 9 × 8 × 7

504

Answer: 504

2
A code uses 3 different numbers chosen from 1 through 11. How many codes are possible?
Show Answer

nPr = n × (n − 1) × … for r factors

11P3 = 11 × 10 × 9

990

Answer: 990

3
A code uses 4 different numbers chosen from 1 through 8. How many codes are possible?
Show Answer

nPr = n × (n − 1) × … for r factors

8P4 = 8 × 7 × 6 × 5

1,680

Answer: 1,680

4
A code uses 4 different numbers chosen from 1 through 10. How many codes are possible?
Show Answer

nPr = n × (n − 1) × … for r factors

10P4 = 10 × 9 × 8 × 7

5,040

Answer: 5,040

5
A code uses 4 different numbers chosen from 1 through 7. How many codes are possible?
Show Answer

nPr = n × (n − 1) × … for r factors

7P4 = 7 × 6 × 5 × 4

840

Answer: 840

Level 3

Hard

Two ranked groups

Example
Gold, silver, and bronze go to the top 3 girls out of 5, and gold, silver, and bronze go to the top 3 boys out of 8. How many ways can the six medals be given?

The places are ranked, so each group is a permutation.

Ways = 5P3 × 8P3

nPr = n × (n − 1) × … for r factors

5P3 = 5 × 4 × 3

60

nPr = n × (n − 1) × … for r factors

8P3 = 8 × 7 × 6

336

60 × 336 = 20,160

Answer: 20,160

1
First and second among 4 flute players, and first, second, and third among 7 drum players. How many ways can the places be filled?
Show Answer

The places are ranked, so each group is a permutation.

Ways = 4P2 × 7P3

nPr = n × (n − 1) × … for r factors

4P2 = 4 × 3

12

nPr = n × (n − 1) × … for r factors

7P3 = 7 × 6 × 5

210

12 × 210 = 2,520

Answer: 2,520

2
Gold and silver among 6 painters, and gold and silver among 5 sculptors. How many ways can the prizes be given?
Show Answer

The places are ranked, so each group is a permutation.

Ways = 6P2 × 5P2

nPr = n × (n − 1) × … for r factors

6P2 = 6 × 5

30

nPr = n × (n − 1) × … for r factors

5P2 = 5 × 4

20

30 × 20 = 600

Answer: 600

3
First, second, and third among 5 soloists, and first and second among 6 understudies. How many ways can the places be filled?
Show Answer

The places are ranked, so each group is a permutation.

Ways = 5P3 × 6P2

nPr = n × (n − 1) × … for r factors

5P3 = 5 × 4 × 3

60

nPr = n × (n − 1) × … for r factors

6P2 = 6 × 5

30

60 × 30 = 1,800

Answer: 1,800

4
Captain and co-captain from 7 seniors, and first, second, and third from 9 juniors. How many ways can the offices be filled?
Show Answer

The places are ranked, so each group is a permutation.

Ways = 7P2 × 9P3

nPr = n × (n − 1) × … for r factors

7P2 = 7 × 6

42

nPr = n × (n − 1) × … for r factors

9P3 = 9 × 8 × 7

504

42 × 504 = 21,168

Answer: 21,168

5
First, second, and third from 4 editors, and first and second from 8 reporters. How many ways can the places be filled?
Show Answer

The places are ranked, so each group is a permutation.

Ways = 4P3 × 8P2

nPr = n × (n − 1) × … for r factors

4P3 = 4 × 3 × 2

24

nPr = n × (n − 1) × … for r factors

8P2 = 8 × 7

56

24 × 56 = 1,344

Answer: 1,344

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