In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?639012645135

The problem involves combinations, a concept in combinatorics. Here's how to solve it:

1. First, we need to select 5 men out of 7. The number of ways to do this is given by the combination formula "n choose r", which is n! / [r!(n-r)!], where n is the total number of options, r is the number of options to choose, and "!" denotes factorial. So, the number of ways to choose 5 men out of 7 is 7! / [5!(7-5)!] = 21.

2. Next, we need to select 2 women out of 3. Using the same formula, the number of ways to do this is 3! / [2!(3-2)!] = 3.

3. Since these are independent events, we multiply the number of ways to get the total number of ways to select a group of 5 men and 2 women out of the total. So, the total number of ways is 21 * 3 = 63.

The problem can be solved using the concept of combinations in mathematics.

Step 1: We need to select 5 men out of 7. The number of ways to do this is given by the combination formula C(n, r) = n! / [(n-r)!r!], where n is the total number of items, and r is the number of items to choose. So, the number of ways to choose 5 men out of 7 is C(7, 5) = 7! / [(7-5)!5!] = 21.

Step 2: We need to select 2 women out of 3. Using the same combination formula, the number of ways to choose 2 women out of 3 is C(3, 2) = 3! / [(3-2)!2!] = 3.

Step 3: Since the selections of men and women are independent events, we multiply the number of ways to get the total number of ways to select a group of 5 men and 2 women out of a total of 7 men and 3 women. So, the total number of ways = 21 * 3 = 63.

The problem can be solved using the concept of combinations in mathematics.

Step 1: We need to select 5 men out of 7. The number of ways to do this is given by the combination formula C(n, r) = n! / [(n-r)!r!], where n is the total number of items, and r is the number of items to choose. So, the number of ways to choose 5 men out of 7 is C(7, 5) = 7! / [(7-5)!5!] = 21.

Step 2: We need to select 2 women out of 3. Using the same combination formula, the number of ways to choose 2 women out of 3 is C(3, 2) = 3! / [(3-2)!2!] = 3.

Step 3: Since the selections of men and women are independent events, we multiply the number of ways to get the total number of ways to select a group of 5 men and 2 women out of a total of 7 men and 3 women. So, the total number of ways is 21 * 3 = 63.