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Chapter 8: Problem 70

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
There are 14400 ways for them to line up with the woman first and alternating thereafter.

Step by step solution

01

Arrangement of the Women

The women always go first, and since they alternate with the men, the women's order will define the men's order. Since there are 5 women, and we are looking at the different ways these 5 women can be arranged, we calculate \(5!\) which equals 120.
02

Arrangement of the Men

By the same logic, since there are 5 men that need to be arranged following the women's order, we calculate \(5!\), which also equals 120.
03

Total Arrangements

Since these arrangements are independent events (arrangement of women and arrangement of men), we multiply the number of arrangements: \(120 (women) * 120 (men)\) equals 14400.

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