A student is allowed to select at most n book | vedclass

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Question

(b) Since the student is allowed to select at most n books out of $(2n + 1)$ books, therefore in order to select one book he has the choice to select

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(b) Since the student is allowed to select at most n books out of $(2n + 1)$ books, therefore in order to select one book he has the choice to select one, two, three, ......, $n$ books.

Thus, if $T$ is the total number of ways of selecting one book then $T = {\,^{2n + 1}}{C_1} + {\,^{2n + 1}}{C_2} + ... + {\,^{2n + 1}}{C_n} = 63$ &hellip;..$(i)$

Again the sum of binomial coefficients

$^{2n + 1}{C_0} + {\,^{2n + 1}}{C_1} + {\,^{2n + 1}}{C_2} + ..... + {\,^{2n + 1}}{C_n} + {\,^{2n + 1}}{C_{n + 1}}$

${ + ^{2n + 1}}{C_{n + 2}} + .... + {\,^{2n + 1}}{C_{2n + 1}} = {(1 + 1)^{2n + 1}} = {2^{2n + 1}}$

or $^{2n + 1}{C_0} + 2{(^{2n + 1}}{C_1} + {\,^{2n + 1}}{C_2} + .. + {\,^{2n + 1}}{C_n}){ + ^{2n + 1}}{C_{2n + 1}} = {2^{2n + 1}}$

==> $1 + 2(T) + 1 = {2^{2n + 1}}$ ==> $1 + T = \frac{{{2^{2n + 1}}}}{2} = {2^{2n}}$

==> $1 + 63 = {2^{2n}}$

$\Rightarrow {2^6} = {2^{2n}}$

$\Rightarrow n = 3$.

A student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select one book is $63$, then the value of $n$ is

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