Question
If ${ }^n C_p={ }^n C_p$ Find ${ }^{12} C_n$.
✓
Answer
We have, If ${ }^n C_p={ }^n C_q=n$
Then $p+q=n$ Also, ${ }^n C_r=\frac{n!}{r!(n-r)!} \ldots$ (i)
$\Rightarrow{ }^n C_4={ }^n C_6 4+6=n $
$\Rightarrow n=10$
Applying (i), ${ }^{12} \mathrm{C}_{10}=\frac{12!}{10!2!}=\frac{12 \times 11 \times 10!}{10!\times 2 \times 1}=\frac{12 \times 11}{2 \times 1}=66$
Then $p+q=n$ Also, ${ }^n C_r=\frac{n!}{r!(n-r)!} \ldots$ (i)
$\Rightarrow{ }^n C_4={ }^n C_6 4+6=n $
$\Rightarrow n=10$
Applying (i), ${ }^{12} \mathrm{C}_{10}=\frac{12!}{10!2!}=\frac{12 \times 11 \times 10!}{10!\times 2 \times 1}=\frac{12 \times 11}{2 \times 1}=66$
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