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Chapter 11: Problem 71

Five men and five women line up at a checkout counter in a store. In how many ways can they line up if the first person in line is a woman, and the people in line alternate woman, man, woman, man, and so on?

Short Answer

Expert verified
The number of ways they can line up is \(5! * 5! = 14,\!400\) ways.

Step by step solution

01

Identify the positions

Based on the constraint, the order of people lining up is: woman, man, woman, man, woman, man, woman, man, woman, and man. Write these down in order. This identifies 10 specific spots (1st, 2nd,3rd, ...,10th) in line to put each individual.
02

Calculate permutations for women

The first, third, fifth, seventh and ninth places are reserved for women. There are 5 women, so the number of possible permutations for the women is 5!, since there are 5 different spots for the 1st woman, 4 for the 2nd woman, 3 for the 3rd woman, 2 for the 4th woman, and 1 for the 5th woman. Calculate 5 factorial (5!) to get these permutations.
03

Calculate permutations for men

The second, fourth, sixth, eighth and tenth places are for men. There are 5 men, so the number of possible permutations for the men is also 5!. Calculate 5 factorial (5!) to get these permutations.
04

Multiply the permutations

Since the men and women have separate spots in the line, you multiply the number of permutations for women and men. The multiplication principle in counting says that if you have a ways of doing one thing, and b ways of doing another, then you have a*b ways of doing both.

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