In how many different ways can a committee consisting of exactly 5 men and 6 women be formed from a group of 8 men and 10 women?

Introduction / Context:
This is a straightforward committee selection problem, where we must choose a fixed number of men and women from separate pools. Order does not matter, so we rely on combinations rather than permutations. The result comes from choosing men and women independently and then multiplying the combination counts to get the total number of committees.

Given Data / Assumptions:
Total men available = 8. Total women available = 10. The committee must include exactly 5 men. The committee must include exactly 6 women. Committee membership is unordered; only who is in the committee matters, not any arrangement.

Concept / Approach:
We treat the selection of men and women as independent tasks. The number of ways to choose 5 men from 8 is 8C5, and the number of ways to choose 6 women from 10 is 10C6. Every selection of men can be combined with every selection of women, so the rule of product tells us to multiply the two combination values. The final answer is 8C5 * 10C6.

Step-by-Step Solution:
Step 1: Compute 8C5 for choosing 5 men from 8. Step 2: Use the symmetry 8C5 = 8C3. Step 3: 8C3 = 8 * 7 * 6 / (3 * 2 * 1) = 56. Step 4: Compute 10C6 for choosing 6 women from 10, using symmetry 10C6 = 10C4. Step 5: 10C4 = 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1) = 210. Step 6: Multiply the two counts: total committees = 56 * 210. Step 7: First compute 56 * 200 = 11200. Step 8: Then add 56 * 10 = 560, giving 11200 + 560 = 11760. Step 9: Therefore, there are 11760 distinct committees possible.

Verification / Alternative check:
We can verify the combination values with factorial formulas: 8C5 = 8! / (5! * 3!) and 10C6 = 10! / (6! * 4!). Calculations will confirm 56 and 210 respectively. Also, note that the size of the committee is 11, and it is reasonable that from a pool of 18 people (8 men plus 10 women) there are many thousands of possible committees. The product 56 * 210 fits this expectation and matches both symmetry and factorial based calculations.

Why Other Options Are Wrong:
The value 53400 is larger and might come from multiplying incorrect combination values. The numbers 17610 and 45000 do not equal 8C5 * 10C6 and reflect arithmetic or formulation errors. The value 60060 is 15C6 and does not correspond to the needed structure. Only 11760 arises from correctly applying the combination formula to both groups and multiplying the results.

Common Pitfalls:
Some students confuse the problem with permutations and try to use nPr instead of nCr, overcounting arrangements where the same group members are listed in different orders. Others may accidentally choose the wrong number of men or women when setting up the combinations. There is also the possibility of incorrectly treating all 18 people as a single pool and using 18C11 without enforcing gender constraints. Carefully separating the selection of men and women prevents these mistakes.

Final Answer:
The number of ways to form the committee is 11760.