The number of committees of size 8 that can be formed from 10 men and 10 women…

Concept:

When a selection must satisfy a minimum-count condition on one group (here, at least 5 women in an 8-member committee), break the problem into separate cases based on how many women are chosen, work out each case as a product of two combinations (nCr for men and nCr for women, chosen independently), and add the cases together since they cannot overlap (the addition rule of counting).

Application:

1. The committee has 8 members chosen from 10 women and 10 men, with the number of women required to be at least 5. Since only 8 seats exist, the number of women can only be 5, 6, 7, or 8.

2. Case (5 women, 3 men): ways = 10C5 × 10C3 = 252 × 120 = 30240

3. Case (6 women, 2 men): ways = 10C6 × 10C2 = 210 × 45 = 9450

4. Case (7 women, 1 man): ways = 10C7 × 10C1 = 120 × 10 = 1200

5. Case (8 women, 0 men): ways = 10C8 × 10C0 = 45 × 1 = 45

6. Total = 30240 + 9450 + 1200 + 45 = 40935

Cross-check:

Every nCr value can be checked with the symmetry rule nCr = nC(n − r): 10C5 = 252, 10C6 = 10C4 = 210, 10C7 = 10C3 = 120, and 10C8 = 10C2 = 45, matching the values used above. Adding the four case totals in a different order — 45 + 1200 = 1245, then 1245 + 9450 = 10695, then 10695 + 30240 = 40935 — gives the same total, confirming the result.

So the number of such committees is 40935.