In how many ways a team of 11 must be selected a team 5 men and 11 women such…

When a selection carries an “at most” restriction on one group, split the count into cases by the exact number taken from that group, count each case with the combinations formula for both groups, and add the cases together (the addition principle for mutually exclusive cases).

Here the team of 11 is drawn from 5 men and 11 women, with the number of men restricted to at most 3 — so the men count can be 0, 1, 2, or 3, and the remaining seats in each case are filled by women.

1. 0 men: 5C0 × 11C11 = 1 × 1 = 1

2. 1 men: 5C1 × 11C10 = 5 × 11 = 55

3. 2 men: 5C2 × 11C9 = 10 × 55 = 550

4. 3 men: 5C3 × 11C8 = 10 × 165 = 1650

5. Adding the four cases: 1 + 55 + 550 + 1650 = 2256

Cross-check by complementary counting: ignoring the men restriction, the team of 11 can be formed from the combined pool of 16 people in 16C11 = 16C5 = 4368 ways. Subtracting the excluded cases with 4 or 5 men — 5C4 × 11C7 = 5 × 330 = 1650 and 5C5 × 11C6 = 1 × 462 = 462, i.e. 2112 in total — gives 4368 − 2112 = 2256, matching the direct case-wise count.

So the number of ways to form the team is 2256.