Question

If the number of 5-element subsets of the set $A=\{a_1,a_2,.....a_{20}\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a_4$, then $k$ is

• A. $5$
• B. $\frac{20}{7}$
• C. $4$
• D. $\frac{10}{3}$

Answer 1

Answer: (C)

Set $A=a_1,a_2,.....a_{20}$ has 20 distinct elements.
We have to select 5-element subset.
$\therefore$ Number of 5-element subsets ${=}^2{C}_5$
According to question
${}^{20}{C}_5=\left({}^{19}{C}_4.k\right)$
$\Rightarrow \frac{20!}{5!15!}=k.\left(\frac{19!}{4!5!}\right)$
$\Rightarrow \frac{20}{5}=k\Rightarrow k=4$