Question

A group contains n men and n women. Identify the steps used to find the number of ways to arrange n men and n women in a row if the men and women alternate?

O There are n men and n women, and all of the Ann = n arrangements are allowed for both men and women.


O There are n men and women, and all of the an. n = 2n1 arrangements are allowed for both men and women. There are exactly two possibilities: elther the row starts with a man and ends with a man, or It starts with a woman and ends with a woman.


O There are exactly two possibilities: either the row starts with a man and ends with a woman, or it starts with a woman and ends with a man


O Arrange the men with women in between them, arrange the women with men in between them, and decide which gender starts the row.


O By the product rule, there are n!-n!-2= 2(n!)² ways

Answer 1

To calculate the number of ways to arrange n men and n women in a row so they alternate, consider two starting cases (man or woman), then apply the factorial to the number of individuals in each group and use the product rule, resulting in 2(n!)^2 total arrangements.

Step-by-step explanation:

To find the number of ways to arrange n men and n women in a row so that they alternate, we would start by considering two cases: either the row starts with a man or it starts with a woman. The total number of arrangements for each case is calculated by the factorial of the number of people of each gender (n! for men and n! for women). Once we arrange either the men or the women, the members of the opposite gender have fixed positions in between. We have two such arrangements since we have two cases (starting with man or woman).

By the product rule in combinatorics, the total number of ways to arrange them would be the product of the number of arrangements for men (n!) and the number of arrangements for women (n!), multiplied by the number of cases (2), which equals 2(n!)^2.