From a total of 7 men and 3 women, in how many different ways can a group of 5 men and 2 women be formed?

Introduction / Context:
This is a straightforward combinations question involving separate selections from two groups. We must form a group with a fixed number of men and a fixed number of women from given pools of each. Because only the composition of the group matters and not the order, combinations are the appropriate tool for counting such groups.

Given Data / Assumptions:

• Total men available = 7.
• Total women available = 3.
• Required group composition: 5 men and 2 women.
• The group is unordered, so the order of members in the group does not matter.

Concept / Approach:
We select 5 men from the 7 using combinations and 2 women from the 3 using combinations. Since these choices are independent and can occur together, the total number of groups is the product of the number of ways to choose the men and the number of ways to choose the women. This is a direct application of the multiplication principle for independent selection stages.

Step-by-Step Solution:
Step 1: Choose 5 men from 7 men. Number of ways = C(7, 5). C(7, 5) = C(7, 2) = 21. Step 2: Choose 2 women from 3 women. Number of ways = C(3, 2) = 3. Step 3: Multiply to get the total number of groups. Total groups = C(7, 5) * C(3, 2) = 21 * 3 = 63.

Verification / Alternative check:
We can verify by noting that for each of the 21 possible selections of 5 men, there are exactly 3 possible pairs of women. So each men selection leads to 3 distinct full groups, and these products do not overlap because the chosen sets of men differ. Thus 21 sets of men each combine with 3 sets of women, giving 21 * 3 = 63 groups in total.

Why Other Options Are Wrong:
Values such as 54 or 64 do not equal the product 21 * 3. The option 36 might come from incorrectly choosing 6 people instead of 7 or misusing the combination formula. The number 70 equals C(8, 2) or C(8, 6) and is unrelated to the specific selection of 5 from 7 and 2 from 3. Only 63 matches the correct combination calculations.

Common Pitfalls:
A common mistake is to try to treat the problem as a single combination from all 10 people, ignoring the requirement that the group must have exactly 5 men and 2 women. Another error is to miscalculate C(7, 5) or C(3, 2). Keeping the male and female selections separate and then multiplying the combination counts helps avoid these errors.

Final Answer:
The number of ways to form a group of 5 men and 2 women is 63.