For a scholarship, at the most n candidates out of 2n + 1 can be selected. If the number of different ways of selection of at least one candidate is 63, the maximum number of candidates that can be selected for the scholarship is

The number of ways you can select at least 1 candidate up to n candidates out of the total 2n+1 is given as 63.

$^{2n+1}C_1+^{2n+1}C_2+...+^{2n+1}C_n$=63 and

$^{2n+1}C_0$+$^{2n+1}C_1$+$^{2n+1}C_2$+...+$^{2n+1}C_n$+$^{2n+1}C_n+1$+$^{2n+1}C_n+2$+...+$^{2n+1}C_2n+1$=$2^{2n+1}$ ..............(1)

We know $^{2n+1}C_0=1$ and $^n{C}_r = ^{n}C_n-r$

So

$^{2n+1}C_0+^{2n+1}C_1+^{2n+1}C_2+...+^{2n+1}C_n=^{2n+1}C_n+1+^{2n+1}Cn+2+...+^{2n+1}C_2n+1$

...........(2)

From(1) and (2)

1+63+63+1=$2^{2n+1}$

$2^{7}$ =$2^{2n+1}$

n=3

Hence,Option(A)3.