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

Introduction / Context:
This question tests your understanding of combinations where you need to choose a specific number of men and women from separate pools. It is a typical selection problem where order does not matter. You are required to use the combination formula separately for men and women and then combine the results using the multiplication principle.

Given Data / Assumptions:
Total number of men = 7. Total number of women = 3. We must form a group with exactly 5 men and exactly 2 women. Order of people in the group does not matter. All individuals are distinct.

Concept / Approach:
We choose men and women independently. The number of ways to choose 5 men from 7 is given by 7C5, and the number of ways to choose 2 women from 3 is given by 3C2. Since each choice of 5 men can be combined with each choice of 2 women, we multiply these two combination values. This is a direct application of the rule of product in combinatorics.

Step-by-Step Solution:
Step 1: Compute the number of ways to choose 5 men from 7: 7C5. Step 2: Use symmetry: 7C5 = 7C2 = 7 * 6 / 2 = 21. Step 3: Compute the number of ways to choose 2 women from 3: 3C2 = 3 * 2 / 2 = 3. Step 4: Multiply the two results to get the total number of groups. Step 5: Total groups = 21 * 3 = 63. Step 6: Therefore, there are 63 different ways to form such a group.

Verification / Alternative check:
Another way to confirm the answer is to list possible compositions logically. There is no flexibility in the counts because the group must have 5 men and 2 women, which means all 3 women are eligible combinations but we only choose 2 at a time. Counting 7C5 directly with factorials as 7! / (5! * 2!) and 3C2 as 3! / (2! * 1!) will still produce 21 and 3 respectively. Multiplying gives 63 again, so the result is consistent under different calculation approaches.

Why Other Options Are Wrong:
Values like 135, 125 and 64 do not match the product 7C5 * 3C2. For example, 7C3 * 3C2 would give 35 * 3 = 105, which still does not appear in the options. The choice 210 might arise from incorrectly choosing all 7 men or mixing counts in the wrong way. Only 63 is directly supported by correct combination formulas for each gender group.

Common Pitfalls:
Common mistakes include using permutations instead of combinations, forgetting that the order of people in the group does not matter, or accidentally choosing the wrong number of men or women. Some students may also incorrectly add combination counts instead of multiplying them, which is incorrect because each combination of men can be paired with each combination of women. Always treat independent selections with the multiplication rule and use nCr for counting choices where order is irrelevant.

Final Answer:
The required group can be formed in 63 different ways.