6.3 Q-14

Exercise 6.3 — Question 14

Problem Statement

If ${}^n{C}_{15}={}^n{C}_7$, find the value of $n$.

Solution

We use the symmetry property of combinations:

${}^n{C}_p={}^n{C}_q⟹p=q\text{or}p+q=n$

Here $p=15$ and $q=7$.

Case 1: $p=q$ $15=7\text{(not possible)}$

Case 2: $p+q=n$ $n=15+7=22$

$\boxed{n=22}$

Key Formula

The symmetry property states: ${}^n{C}_r={}^n{C}_{n-r}$

This means ${}^n{C}_p={}^n{C}_q$ when $q=n-p$, i.e., $p+q=n$.