It's a committee → use C, not P: A committee/group has no order — {Ana, Ben} is the same committee as {Ben, Ana} — so you CHOOSE with ⁿCᵣ, never arrange with ⁿPᵣ. (Use P only when there are positions/ranks/roles.)
Need exactly so many from each type (say 2 men AND 2 women)? Choose from each group, then multiply (AND → ×).
IB-style question — exactly 2 men
A committee of 4 is chosen from 5 men and 6 women. It must contain exactly 2 men.
How many committees are possible?
Step by step
- The committee needs 2 men AND 2 women (that's 4). Choose the men.
- Choose the women.
- Each of the 10 ways to pick the men can pair with ANY of the 15 ways to pick the women — that's why 'AND' means multiply.
Final answer
150 committees.
A committee has no order, so we CHOOSE each group (C, not P): ⁵C₂ × ⁶C₂ = 10 × 15 = 150.
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Add the cases (or take away the unwanted): "At least 2 women" = exactly 2 + exactly 3 + … — add the cases.
If that's a lot of cases, do total − the unwanted instead.
IB-style question — at least 2 women
A team of 4 is chosen from 5 men and 6 women. It must contain at least 2 women.
How many teams?
Step by step
- Exactly 2 women (2W, 2M).
- Exactly 3 women (3W, 1M).
- Exactly 4 women (4W, 0M).
- Add the cases.
Final answer
265 teams.
Faster: count the opposite: Total minus the unwanted (0 or 1 woman):
¹¹C₄ − (⁵C₄ + ⁶C₁×⁵C₃) = 330 − (5 + 60) = 265 — same answer.