If ^nC_p = ^nC_p Find ^ 12 C_n. - Vidyadip

Question

If $^nC_p =\ ^nC_p$ Find $^{12}C_n.$

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Answer

We have,
If $^nC_p =\ ^nC_q = n$
Then $p + q = n$
Also,
${^\text{n}}\text{C}_{\text{r}}=\frac{\text{n!}}{\text{r!}(\text{n}-\text{r})!}\ ...(\text{i})$
$\Rightarrow\ ^nC_4 =\ ^nC_6$
$4 + 6 = n$
$\Rightarrow n = 10$
Applying (i),
${^\text{12}}\text{C}_{\text{10}}=\frac{\text{12!}}{10!2!}$
$=\frac{12\times11\times10!}{10!\times2\times1}$
$=\frac{12\times11}{2\times1}=66$

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