If nCp = nCp Find 12Cn. - Vidyadip

Question

If ⁿC_p = ⁿC_p Find ¹²C_n.

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Answer

We have,
If ⁿC_p = ⁿC_q = n
Then p + q = n
Also,
${^\text{n}}\text{C}_{\text{r}}=\frac{\text{n!}}{\text{r!}(\text{n}-\text{r})!}\ ...(\text{i})$
⇒ ⁿC₄ = ⁿC₆
4 + 6 = n
⇒ 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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