6.3 Q-7

Exercise 6.3 — Question 7

Problem Statement

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

Key Property (Symmetry of Combinations)

For combinations, the following symmetry property holds:

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

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

Solution

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

Applying the symmetry property:

Case 1: $15=7$ — This is impossible.

Case 2: $15+7=n$

$n=22$

Verification

${}^{22}{C}_{15}={}^{22}{C}_7=\frac{22!}{7!\cdot 15!}=170544✓$

Real-World Connection

The symmetry property of combinations has practical applications. For example, choosing 7 items to include from 22 is equivalent to choosing 15 items to exclude — both describe the same selection. This principle is used in lottery probability calculations, where choosing $r$ winners from $n$ candidates is the same as choosing $n-r$ non-winners.