Question

From a group of persons the number of ways of selecting 5 persons is equal to that of 8 persons. The number of persons in the group is
(a) 13
(b) 40
(c) 18
(d) 21

Answer 1

Hint: We have to assume the number of persons in the group as n. Then, the number of ways of selecting 5 persons from n persons will be $^{n}{{C}_{5}}$ and the number of ways of selecting 8 persons from n persons will be $^{n}{{C}_{8}}$ . Now, according to the given condition, we will equate $^{n}{{C}_{5}}$ and $^{n}{{C}_{8}}$ . Then, we have to use the property that if $^{n}{{C}_{k}}{{=}^{n}}{{C}_{r}}$ , then either $k=r$ or \[n=k+r\] . Then, we have to find the value of x.

Complete step by step answer:
We have to find the number of persons in the group. Let us assume the number of persons in the group as n. We know that the number of ways of choosing r persons from n persons is given by $^{n}{{C}_{r}}$ .
We are given that the number of ways of selecting 5 persons from a group of n persons is equal to that of 8 persons. We can write the number of ways of selecting 5 persons from n persons as $^{n}{{C}_{5}}$ and the number of ways of selecting 8 persons from n persons as $^{n}{{C}_{8}}$ . Then, according to the given condition,
$^{n}{{C}_{5}}{{=}^{n}}{{C}_{8}}$
We know that if $^{n}{{C}_{k}}{{=}^{n}}{{C}_{r}}$ , then either $k=r$ or \[n=k+r\] . Here, we know that 5 is not equal to 8. So we can go for \[n=k+r\] .
$\begin{align}
  & ^{n}{{C}_{5}}{{=}^{n}}{{C}_{8}} \\
 & \Rightarrow n=5+8 \\
\end{align}$
Let us add 5 and 8.
$\Rightarrow n=13$

So, the correct answer is “Option a”.

Note: We have used combination here instead of permutation because combinations are used when the order doesn’t matter. We use permutation when order matters. Students must be thorough with the formulas and properties of combinations. Here, we did not expand the combinations. Instead, we have applied the property. Expanding the equation $^{n}{{C}_{5}}{{=}^{n}}{{C}_{8}}$ and solving for n will be a time consuming task.