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If the number of 5-element subsets of the set $A=\left\{a_1, a_2, \ldots, a_{20}\right\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a_4$, then $k$ is
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The correct answer is:
4
4
Set $A=\left\{a_1, a_2, \ldots . ., a_{20}\right\}$ has 20 distinct elements.
We have to select 5-element subset.
$\therefore$ Number of 5-element subsets $={ }^{20} \mathrm{C}_5$
According to question
$$
\begin{aligned}
& { }^{20} \mathrm{C}_5=\left({ }^{19} \mathrm{C}_4 \cdot k\right. \\
& \Rightarrow \frac{20 !}{5 ! 15 !}=k \cdot\left(\frac{19 !}{4 ! 15 !}\right) \\
& \Rightarrow \quad \frac{20}{5}=k \Rightarrow k=4
\end{aligned}
$$
We have to select 5-element subset.
$\therefore$ Number of 5-element subsets $={ }^{20} \mathrm{C}_5$
According to question
$$
\begin{aligned}
& { }^{20} \mathrm{C}_5=\left({ }^{19} \mathrm{C}_4 \cdot k\right. \\
& \Rightarrow \frac{20 !}{5 ! 15 !}=k \cdot\left(\frac{19 !}{4 ! 15 !}\right) \\
& \Rightarrow \quad \frac{20}{5}=k \Rightarrow k=4
\end{aligned}
$$
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