Practice 2 Solutions

1. $5!$

$5·4·3·2·1$

150

2. $nPr(5, 3)$

$$\begin{gathered} \frac{5!}{\left(5–3\right)!} \\ \frac{5!}{2!} \\ \frac{5·4·3·\cancel{2!}}{\cancel{2!}} \end{gathered}$$

60

3. $nCr(20, 18)$

$$\begin{gathered} \frac{20!}{18!\left(20–18\right)!} \\ \frac{20!}{18!2!} \\ \frac{20·19·\cancel{18!}}{\cancel{18!}2!} \\ \frac{20·19}{2·1} \end{gathered}$$

190

4. $3!$

$3·2·1$

6

5. $nPr(7, 2)$

$$\begin{gathered} \frac{7!}{\left(7–2\right)!} \\ \frac{7!}{5!} \\ \frac{7·6·\cancel{5!}}{\cancel{5!}} \end{gathered}$$

42

6. $nCr(10, 8)$

$$\begin{gathered} \frac{10!}{8!\left(10–8\right)!} \\ \frac{10!}{8!2!} \\ \frac{10·9·\cancel{8!}}{\cancel{8! }2!} \end{gathered}$$

45

7. Determine how many ways you can arrange the letters MATH. None of the arrangements needs to form a new word.

$\begin{array}{cccc}\underline{ 4 }& \underline{ 3 }& \underline{ 2 }& \underline{ 1 }\\ n& \left(n–1\right)& \left(n–2\right)& 1\end{array}$$4!=4·3·2·1=24$

The letters in MATH can be arranged in 24 different ways.

8. The Billie Book Shoppe has a “Buy 3, Get 1 Free” deal from a selection of 20. How many different combinations of four books can you choose?

$$\begin{gathered} nCr\left(20, 4\right) \\ \frac{20!}{4!\left(20–4\right)!} \\ \frac{20!}{4!16!} \\ \frac{20·19·18·17·\cancel{16!}}{\cancel{16!} 4·3·2·1} \end{gathered}$$

4,845 book combinations can be chosen.

9. The Department of Motor Vehicles requires license plates on all vehicles. How many unique license plates can be made if a three-letter, three-number combination is used?

${26}^3·{10}^3$

17,576,000 unique license plates can be made.

10. A team of scientists programmed a boogie robot. The dancing robot knows 17 dance moves. To submit the robot for a competition, a routine with six dance moves must be prepared. How many ways can the boogie robot (get down) be programmed?

$$\begin{gathered} nPr\left(17, 6\right) \\ \frac{17!}{\left(17–6\right)!} \\ \frac{17!}{11!} \\ \frac{17·16·15·14·13·12·\cancel{11!}}{\cancel{11!}} \end{gathered}$$

The scientists can program the robot 8,910,720 different ways.

11. Pettigrew’s Petals keeps fifteen different types of flowers in stock for their arrangements. If they guarantee every arrangement will have eight types of flowers, how many flower arrangements can be created?

$$\begin{gathered} nCr\left(15, 8\right) \\ \frac{15!}{8!\left(15–8\right)!} \\ \frac{15!}{8!7!} \\ \frac{\cancel{15}·\cancel{14}·13·12·11·10·9·\cancel{8!}}{\cancel{8!} \cancel{7}·6·\cancel{5}·4·\cancel{3}·\cancel{2}·1} \\ \frac{13·12·11·10·9}{6·4·1} \end{gathered}$$

Pettigrew’s Petals can create 6,435 flower arrangements.

12. There are 12 players on a basketball team, but only 5 will be on the court at the same time, each in a unique position. How many ways can the coach set the lineup for the specific positions?

$$\begin{gathered} nPr\left(12, 5\right) \\ \frac{12!}{\left(12–5\right)!} \\ \frac{12!}{7!} \\ \frac{12·11·10·9·8·\cancel{7!}}{\cancel{7!}} \end{gathered}$$

The coach can create 95,040 team lineups for the basketball team.

13. The tenth-grade student council at Stars Hollow High has ten members. They need a committee of four students to plan the upcoming Spring Fling dance. How many different committees are possible?

$$\begin{gathered} nCr\left(10, 4\right) \\ \frac{10!}{4!\left(10–4\right)!} \\ \frac{10!}{4!6!} \\ \frac{10·9·8·7·\cancel{6!}}{4! \cancel{6!}} \\ \frac{10·{\displaystyle \stackrel{3}{\cancel{9}}}·\cancel{8}·7}{\cancel{4}·\cancel{3}·\cancel{2}·1} \\ 10·3·7 \end{gathered}$$

Stars Hollow High can create 210 different committees of tenth-grade student council members.

14. Luke’s Diner offers 5 types of bread, 7 types of meat, 4 cheeses, and 8 toppings. How many different sandwiches can you create if you choose one type of bread, one meat, one cheese, and two toppings?

$5·7·4·8·7$

Luke’s Diner can create 7,840 sandwiches.

15. Picking a color palette to decorate a room

This is a combination because the order in which you pick the colors does not change the final look of the room.

16. Creating a physical therapy plan after surgery with numbered steps

This is a permutation because physical therapy is completed in a specific order (numbered steps).