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(B) The number of ways to select at least one candidate is given by $\sum_{k=1}^{n} {^{2n+1}C_k} = 63$. We know that $\sum_{k=0}^{2n+1} {^{2n+1}C_k} =

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$3$

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(B) The number of ways to select at least one candidate is given by $\sum_{k=1}^{n} {^{2n+1}C_k} = 63$. We know that $\sum_{k=0}^{2n+1} {^{2n+1}C_k} = 2^{2n+1}$. Since ${^{2n+1}C_k} = {^{2n+1}C_{2n+1-k}}$,we have $\sum_{k=0}^{n} {^{2n+1}C_k} = \sum_{k=n+1}^{2n+1} {^{2n+1}C_k} = \frac{2^{2n+1}}{2} = 2^{2n}$. Thus,$\sum_{k=1}^{n} {^{2n+1}C_k} = (\sum_{k=0}^{n} {^{2n+1}C_k}) - {^{2n+1}C_0} = 2^{2n} - 1$. Given $2^{2n} - 1 = 63$,we get $2^{2n} = 64 = 2^6$. Therefore,$2n = 6$,which implies $n = 3$. The maximum number of candidates that can be selected is $n = 3$.

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

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