Problem 70 - College Algebra

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.