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(D) The maximum value of ${}^6C_r$ occurs at $r = \frac{6}{2} = 3$. The value is ${}^6C_3 = \frac{6!}{3!3!} = \frac{6 	imes 5 	imes 4}{3 	imes 2

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$4$

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(D) The maximum value of ${}^6C_r$ occurs at $r = \frac{6}{2} = 3$. The value is ${}^6C_3 = \frac{6!}{3!3!} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$. Given that the difference between the maximum value of ${}^6C_r$ and ${}^nC_3$ is $16$,we have $|20 - {}^nC_3| = 16$. This implies ${}^nC_3 = 20 + 16 = 36$ or ${}^nC_3 = 20 - 16 = 4$. Case $1$: ${}^nC_3 = 36$ $\Rightarrow \frac{n(n-1)(n-2)}{6} = 36$ $\Rightarrow n(n-1)(n-2) = 216$. There is no integer $n$ satisfying this. Case $2$: ${}^nC_3 = 4$ $\Rightarrow \frac{n(n-1)(n-2)}{6} = 4$ $\Rightarrow n(n-1)(n-2) = 24$. Testing values,for $n=4$,$4 	imes 3 	imes 2 = 24$. Thus,$n = 4$.

The difference between the maximum value of ${}^6C_r$ and ${}^nC_3$ is $16$. Then $n=$

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