If there are n men and n women in the group, we can first arrange the men and women separately in alternate positions. We can treat the men as a single group and arrange them in n! ways, and similarly, we can treat the women as a single group and arrange them in n! ways.
Next, we need to arrange the two groups (men and women) in alternate positions to satisfy the given condition. Since we need to place the youngest person (who could be a man or a woman) at the beginning of the row, we have two cases:
Case 1: The youngest person is a man
In this case, the men must start the arrangement, followed by the women. Since there are n men and n women, there are n! ways to arrange the men and n! ways to arrange the women. We can fix the youngest man at the beginning of the row, so there are (n-1)! ways to arrange the remaining n-1 men. Similarly, there are (n-1)! ways to arrange the n-1 women. Therefore, the total number of arrangements in this case is:
n! * n! * (n-1)! * (n-1)!
Case 2: The youngest person is a woman
In this case, the women must start the arrangement, followed by the men. Since there are n women and n men, there are n! ways to arrange the women and n! ways to arrange the men. We can fix the youngest woman at the beginning of the row, so there are (n-1)! ways to arrange the remaining n-1 women. Similarly, there are (n-1)! ways to arrange the n-1 men. Therefore, the total number of arrangements in this case is:
n! * n! * (n-1)! * (n-1)!
To get the total number of arrangements that satisfy the given condition, we need to add the number of arrangements in Case 1 and Case 2:
n! * n! * (n-1)! * (n-1)! + n! * n! * (n-1)! * (n-1)!
= 2 * n! * n! * (n-1)! * (n-1)!
= 2 * (n!)^2 * (n-1)!
Therefore, the total number of ways to arrange the people in a row with alternating men and women, starting with the youngest person, is 2 * (n!)^2 * (n-1)!.