If the number of 5-element subsets of the set A=a_1,,a_20 of 20 distinct elements is k tim…

Question

If the number of 5-element subsets of the set $A=\{a_1,\ldots,a_{20}\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a_4$, then $k$ is:

Accepted Answer

Total 5-element subsets: $\binom{20}{5}$. Subsets containing $a_4$: $\binom{19}{4}$ (choose remaining 4 from the other 19).

$$\binom{20}{5} = k\cdot\binom{19}{4} \implies \frac{20!}{5!15!} = k\cdot\frac{19!}{4!15!} \implies \frac{20}{5}=k \implies k=4$$

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