There are 6 men and 11 women. In how many ways can one select a group of 6…

Concept: For a selection with a minimum-count restriction ("at least k men"), split the selection into mutually exclusive cases by the number of men chosen, evaluate each case with the combination formula nCr = n! / (r!(n − r)!), then add the case totals together (the addition principle for mutually exclusive, non-overlapping cases).

Step-by-step: Split the group of 6 by the number of men, since 6 available men and 11 available women cap what each case can hold.

1. Exactly 3 men and 3 women: 6C3 × 11C3 = 20 × 165 = 3300

2. Exactly 4 men and 2 women: 6C4 × 11C2 = 15 × 55 = 825

3. Exactly 5 men and 1 woman: 6C5 × 11C1 = 6 × 11 = 66

4. Exactly 6 men and 0 women: 6C6 × 11C0 = 1 × 1 = 1

5. Total = 3300 + 825 + 66 + 1 = 4192

Cross-check: Choosing any 6 people from all 17 (6 + 11) can be done in 17C6 = 12376 ways. Subtract the cases with fewer than 3 men (0, 1, or 2 men): 6C0×11C6 + 6C1×11C5 + 6C2×11C4 = 462 + 2772 + 4950 = 8184. So 12376 − 8184 = 4192, matching the total above.