6.3 Q-6

Exercise 6.3 — Question 6

Problem

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

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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 ${}^n{C}_r={}^n{C}_{n-r}$.

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Solution

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

Applying the symmetry property:

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

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

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Verification

${}^{22}{C}_{15}={}^{22}{C}_7✓$

This holds because ${}^{22}{C}_{15}={}^{22}{C}_{22-15}={}^{22}{C}_7$.