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 $=^2C_5$
According to question
$^{20}C_{5}=\left(^{19}C_{4}.k\right)$
$\Rightarrow \frac{20!}{5!\,15!}=k. \left(\frac{19!}{4!\,5!}\right)$
$\Rightarrow \frac{20}{5}=k \Rightarrow k=4$