6.3 Q-9

Exercise 6.3 — Question 9

Problem

Solve for $n$ if ${}^n{C}_{15}={}^n{C}_7$.

Key Property

The symmetry property of combinations states:

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

This follows directly from the formula ${}^n{C}_r=\frac{n!}{r!(n-r)!}$, since ${}^n{C}_r={}^n{C}_{n-r}$.

Solution

Given: ${}^n{C}_{15}={}^n{C}_7$

Using the symmetry property:

Case 1: $p=q$ $15=7\text{(impossible)}$

Case 2: $p+q=n$ $15+7=n$ $\boxed{n=22}$

Verification

${}^{22}{C}_{15}={}^{22}{C}_7=\frac{22!}{7!\cdot 15!}✓$