91Ó°ÊÓ

First, we will find the number of ways to choose 3 men from a group of 10 men, which is a combination problem. We can use the combination formula as follows: C(n, k) = n! / (k!(n-k)!), where n is the total number of items and k is the number of items to be selected, and ! stands for the factorial of a number. For this case, n = 10 (total men) and k = 3 (number of men to be selected). Number of ways to choose 3 men = C(10, 3) = 10! / (3! * (10-3)!) After selecting 3 men, there are two more positions to be filled in the committee. Since the committee must have 3 men, these two positions must be filled by women. So, now we need to find the number of ways to choose 2 women from the group of 6 women. For this case, n = 6 (total women) and k = 2 (number of women to be selected). Number of ways to choose 2 women = C(6, 2) = 6! / (2! * (6-2)!) Now, to find the total number of ways to have a committee with exactly 3 men, we multiply the number of ways to select 3 men by the number of ways to select 2 women (as the choices are independent and can be combined in every possible way). Total committees with 3 men = (number of ways to choose 3 men) * (number of ways to choose 2 women)

For this case, we will consider two possible scenarios: 1. A committee with exactly 3 men (which we have already calculated in the previous case) 2. A committee with exactly 4 men For the second scenario, we will find the number of ways to choose 4 men from a group of 10 men. For this case, n = 10 (total men) and k = 4 (number of men to be selected). Number of ways to choose 4 men = C(10, 4) = 10! / (4! * (10-4)!) After selecting 4 men, there is one more position to be filled in the committee. Since the committee must have at least 3 men, this last position must be filled by a woman. So, now we need to find the number of ways to choose 1 woman from the group of 6 women. For this case, n = 6 (total women) and k = 1 (number of women to be selected). Number of ways to choose 1 woman = C(6, 1) = 6! / (1! * (6-1)!) Now, to find the total number of ways to have a committee with at least 3 men, we need to add the number of possible committees with exactly 3 men to the number of possible committees with exactly 4 men. Total committees with at least 3 men = (number of ways to choose 3 men and 2 women) + (number of ways to choose 4 men and 1 woman)