If ^n C_ 12 = ^n C_ 5 , Find the value of n.

Question

If ${^\text{n}}\text{C}_{12}={^\text{n}}\text{C}_{5},$ Find the value of $n.$

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Answer

We have,
${^\text{n}}\text{C}_{\text{r}}=\frac{\text{n!}}{\text{r!}(\text{n}-\text{r})!}$
Hence, $n = m$
$r = 12$ and $5$
Applying formula
$^nC_p =\ ^nC_q = n$
Then$ p + q = n$
$\Rightarrow ^nC_{12} =\ ^nC_5$
$12 + 5 = n$
$\Rightarrow n = 17$

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