Exercise 187 Page 664

a Given that Kristen gets her aisle seat, there are 7 seats left once she has her pick. There are two aisle seats, which means she has two choices of where to sit.

This can be interprested as a permutation where Kristen selects 1 chair from a set of 2 chairs.

_2P_1=2!/1!=2! When Kristen has picked a seat we have seven seats left. Notice that the order in which the friends sit matters. As each person picks a seat, the number of seats subtracts by 1. Therefore, the number of arrangements can be expressed as 7! after Kristen has made her choice.

The total number of arrangements is the product of 2! and 7!. Let's calculate this. 2!* 7!=10 080

b In this situation we want to divide the number of arrangements where we have three girls on the right-hand side and five boys on the left-hand side with the total number of arrangements if the group of friends choose seats randomly.

Total Number of Arrangements

The total number of arrangements, without regard to which gender sits where, is _8P_8. _8P_8=8!/( 8- 8)! ⇔ _8P_8=40 320

Arrangements of 5 Boys, 3 Girls

Next, we will find the number of arrangements where 3 girls occupy the three seats on the row's right-hand side. This can be expressed as _3P_3. _3P_3=3!/( 3- 3)! ⇔ _3P_3=6 In the same way, we can find the number of arrangements with the 5 boys on the left-hand side. _5P_5=5!/( 5- 5)! ⇔ _5P_5=120 The total number of arrangements, where the girls are on the right-hand side and the boys on the left-hand side, is the product of the number of ways the girls can be arranged and the number of ways the boys can be arranged. If we divide this by the total arrangements, we can calculate the probability. _3P_3* _5P_5/_8P_8 =6* 120/40 320≈ 1.8 %

c We have a group of 8 and want to know how many ways they can be arranged. Notice that this is a permutation where some of the last name's first letters repeat. Therefore, we want to use the following formula.

n!/r_1!r_2!...

In this formula, n is the number of items and r_1, r_2,... is the number of times that item 1, item 2, and so on repeats. As already explained there are 8 names, which means n= 8. Also, there are 2 names that begin with K and 3 that begins with S. This means we have r_1= 2. and r_2= 3. 8!/2! 3! Let's calculate this on a graphing calculator.