combinations

Question

If C(n, 4) = C(n, 6), then n equals

RankGuru · Accepted Answer

1. **Property of Combinations**: $\binom{n}{r} = \binom{n}{n-r}$.
2. **Apply Property**: This implies that if $\binom{n}{r} = \binom{n}{k}$, then either $r = k$ or $r + k = n$.
3. **Solve for $n$**:
   - Since $4 
eq 6$, it must be that $4 + 6 = n$.
   - Therefore, $n = 10$.
4. **Verification**: $\binom{10}{4} = \frac{10!}{4!6!}$ and $\binom{10}{6} = \frac{10!}{6!4!}$. They are identical.

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