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Chapter 14: Problem 19

If three boys and three girls sit in a row on a park bench, and no boy can sit on either end of the bench, how many arrangements of the children on the bench are possible? a. 46,656 b. 38,880 c. 1,256 d. 144 e. 38

Short Answer

Expert verified
There are 144 possible arrangements, meaning the correct choice is (d) 144.

Step by step solution

01

Positioning the Girls

Since no boy can sit on either end of the bench, it's a given that girls will be positioned at both ends. In addition, one girl will be in the middle. Therefore, there are 4 possible spaces where the girls and boys can sit: _ G _, _ G _, _ G _. The girls can be arranged in these three chosen spots in 3! (3-factorial) ways.
02

Positioning the Boys

With the girls positioned, there are essentially 4 slots remaining where the boys can sit, designated by the underscores in the girl arrangement. These are equivalent to the spaces between the girls and at the ends of the bench. The boys can be arranged in these four spots in 4P3 (Permutation of 4 slots taken 3 at a time) ways.
03

Calculating the total arrangements

The total number of arrangements is obtained by multiplying the number of ways the girls can arrange and the number of ways the boys once the girls have been seated. Thus it is 3! * 4P3.
04

Numerical computation

Calculate 3!(=6) * 4P3 [=4!/(4-3)!] (=24), which equals to 144.

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