A group consists of 7 men and 5 women. In how many ways can a group of 4 members be selected if the group has no women?
(A) 70
(B) 30
(C) 24
(D) 35

Question

Accepted Answer

> [!TLDR]
> A committee with no women is drawn entirely from the 7 men: $\binom{7}{4} = 35$.

### Step 1 — Reduce the pool
If the group of 4 can have no women, the 5 women are excluded; all 4 come from the 7 men. Selection (not arrangement), so use combinations:

> [!FORMULA]
> $$\binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$

### Step 2 — Apply with $n=7$, $r=4$
Use $\binom{7}{4}=\binom{7}{3}$ for an easier computation:

$$\binom{7}{4} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = \frac{210}{6} = 35$$ …

Combinations (nCr)

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Step 1 — Reduce the pool

Step 2 — Apply with $n=7$ , $r=4$

Step 1 — Reduce the pool

Step 2 — Apply with n=7n=7n=7, r=4r=4r=4

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Step 2 — Apply with $n=7$ , $r=4$

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