Question
How many ways can a committee of 5 be formed from 7 men and 6 women, if the committee must include at least 2 women?
Solution — Step by Step
“At least 2 women” means: 2 women + 3 men, 3 women + 2 men, 4 women + 1 man, or 5 women + 0 men.
- 2W + 3M:
- 3W + 2M:
- 4W + 1M:
- 5W + 0M:
Final answer: committees.
Why This Works
“At least” or “at most” problems split into mutually exclusive cases. Add the case counts (since the cases don’t overlap). Each case uses the multiplication principle: choose women × choose men.
Alternative Method (Complement)
Total committees of 5 from 13: .
Subtract cases with fewer than 2 women:
- 0 women, 5 men:
- 1 woman, 4 men:
. Same answer, often faster when “at least” includes most cases.
For “at least of something” with many cases, use the complement: total minus (cases with fewer than ). For “at least ” with few excluded cases, this is much faster than enumerating.
Students forget the case “5 women, 0 men” thinking the committee must have at least one man. The problem says “at least 2 women” with no constraint on men, so all-women committees count. Read the constraint carefully.