For a scholarship, atmost n candidates out of 2n+1 can be selected. If number of different ways of s

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6.Permutation and Combination

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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 -

$2$

$3$

$4$

$5$

Solution

$ ^{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 $
$ \therefore {2^{2n}} – 1 = 63 \Rightarrow {2^{2n}} = 64 = {2^6} \hfill $
$\Rightarrow n = 3 \hfill]$

Standard 11

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Solution

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