If ^n C_ 10 = ^n C_ 12 , Find ^ 23 C_n. - Vidyadip

Question

If ${ }^n C_{10}={ }^n C_{12}$, Find ${ }^{23} C_n$.

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Answer

We have, If ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{p}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{q}}=\mathrm{n}$
Then $\mathrm{p}+\mathrm{q}=\mathrm{n}$
Also, $\Rightarrow{ }^{\mathrm{n}} \mathrm{C}_{10}={ }^{\mathrm{n}} \mathrm{C}_{12} 10+12=\mathrm{n} \Rightarrow \mathrm{n}=22$
Applying (i), ${ }^{23} \mathrm{C}_{22}=\frac{23!}{22!1!}$
$=\frac{23 \times 22!}{22!}=23$

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