6.3 Q-8

Exercise 6.3 — Question 8

Problem

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

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Key Concept: Symmetry Property of Combinations

For any combination ${}^n{C}_p={}^n{C}_q$, either:

$p=q\text{or}p+q=n$

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: $15=7$ — This is impossible.

Case 2: $15+7=n$

$n=22$

Verification: ${}^{22}{C}_{15}={}^{22}{C}_7✓(\text{since }{}^n{C}_r={}^n{C}_{n-r})$

$\boxed{n=22}$

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