For a scholarship, atmost n candidates out of | vedclass

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Question

$  ^{2n + 1}{C_1}{ + ^{2n + 1}}{C_2} + ......{ + ^{2n + 1}}{C_n} = 63. \hfill $
  Also hfill 
 $ ^{2n + 1}{C_0}{ + ^{2n + 1}}{C_

Vedclass · Accepted Answer

$3$

Vedclass · Answer

$  ^{2n + 1}{C_1}{ + ^{2n + 1}}{C_2} + ......{ + ^{2n + 1}}{C_n} = 63. \hfill $
  Also hfill 
 $ ^{2n + 1}{C_0}{ + ^{2n + 1}}{C_1} + ......{ + ^{2n + 1}}{C_n}{ + ^{2n + 1}}{C_{n + 1}} + .....{ + ^{2n + 1}}{C_{2n + 1}} = {2^{2n + 1}} \hfill $
$   \Rightarrow 2 + 2{(^{2n + 1}}{C_1}{ + ^{2n + 1}}{C_2} + .....{ + ^{2n + 1}}{C_n}) = {2^{2n + 1}} \hfill $
 $ { \Rightarrow ^{2n + 1}}{C_1}{ + ^{2n + 1}}{C_2} + .....{ + ^{2n + 1}}{C_n} = {2^{2n}} - 1 \hfill $
$  	herefore {2^{2n}} - 1 = 63 \Rightarrow {2^{2n}} = 64 = {2^6} \hfill $
   $\Rightarrow n = 3 \hfill]$

For a scholarship, atmost $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selection of atleast one candidate for scholarship is $63$, then maximum number of candidates that can be selected for the scholarship is -

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