Practice 1 Solutions

1. $4!$

$4·3·2·1$

24

2. $nPr(4, 4)$

$$\begin{gathered} \frac{4!}{\left(4–4\right)!}=\frac{4!}{0!} \\ \frac{4·3·2·1}{1} \end{gathered}$$

24

3. $nCr(8, 6)$

$$\begin{gathered} \frac{8!}{6!\left(8–6\right)!} \\ \frac{8!}{6!2!} \\ \frac{8·7·\cancel{6!}}{\cancel{6!}2!} \\ \frac{8·7}{2·1} \end{gathered}$$

28

4. $nCr(4, 4)$

$$\begin{gathered} \frac{4!}{4!\left(4–4\right)!} \\ \frac{4!}{4!} \end{gathered}$$

1

5. $6!$

$6·5·4·3·2·1$

720

6. $nPr(6, 2)$

$$\begin{gathered} \frac{6!}{\left(6–2\right)!} \\ \frac{6!}{4!} \\ \frac{6·5·\cancel{4!}}{\cancel{4!}} \end{gathered}$$

30

7. $7!$

$7·6·5·4·3·2·1$

5,040

8. $12!$

$12·11·10·9·8·7·6·5·4·3·2·1$

479,001,600

9. In the 1800s, maritime vessels could display flags to send messages to other ships at sea. A ship would usually have 40 flags and might display five at a time. How many messages could a ship display?

$$\begin{gathered} nPr\left(40, 5\right) \\ \frac{40!}{\left(40–5\right)!} \\ \frac{40·39·38·37·36·\cancel{35!}}{\cancel{35!}} \end{gathered}$$

Maritime vessels could display 78,960,960 unique messages in the 1800s.

10. How many unique 4-letter codes can you create using the 26 letters of the English alphabet?

${26}^4$

Using the 26 letters of the alphabet, 456,976 unique 4-letter codes can be created.

11. A jury of 12 people will be selected from a group of 50 for a trial. Each member of the jury has the same roles and responsibilities. How many combinations of juries can be selected?

$$\begin{gathered} nCr\left(50, 12\right) \\ \frac{50!}{12!\left(50–12\right)!} \\ \frac{50!}{12!38!} \\ \frac{50·49·48·47·46·45·44·43·42·41·40·39·\cancel{38!}}{\cancel{38! }12·11·10·9·8·7·6·5·4·3·2·1} \end{gathered}$$

$1.21\times {10}^{11}$ combinations of juries can be selected.

12. Gerry is making granola for breakfast. She has two types of oats, five types of dried fruit, and three types of seeds. How many different granola recipes can she create?

$2·5·3$

Gerry can create 30 types of granola.

13. Something About Salad offers thirteen ingredients to customize a salad. Customers can select five ingredients to create their salads. How many different salads can be created?

$$\begin{gathered} nCr\left(13, 5\right) \\ \frac{13!}{5!\left(13–5\right)!} \\ \frac{13!}{5!8!} \\ \frac{13·\cancel{12}·11·\cancel{10}·9·\cancel{8!}}{\cancel{8!} \cancel{5}·\cancel{4·3}·\cancel{2}·1} \\ \frac{13·11·8}{1} \end{gathered}$$

Something About Salad customers can create 1,287 different salads.

14. The finals of the 100-meter dash consist of eight runners. Gold, silver, and bronze medals are awarded for first, second, and third place. How many ways can the runners place?

$$\begin{gathered} nPr\left(8, 3\right) \\ \frac{8!}{\left(8–3\right)!} \\ \frac{8!}{5!} \\ \frac{8·7·6·\cancel{5!}}{\cancel{5!}} \end{gathered}$$

There are 336 ways the runners can finish gold, silver, and bronze.

15. Playing notes of a song on an instrument

This is a permutation because the order of the notes matters.

16. Selecting coffee beans for your own signature blend

This is a combination because the order in which you select the coffee beans does not change the blend.