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 the books is 63, then n is :

Explanation

Given student select atlest one and atmost n books from a collectionn of (2n+1) books. It means that he select one book or two books or three books or . . . Or n books.
Hence, by the given hypothesis.
`.^(2n+1)C_(1)+.^(2n+1)C_(2)+.^(2n+1)C_(3)+. . . .+.^(2n+1)C_(n)=63` . . . (i)
Also, the sum of binomial coefficients, is
`.^(2n+1)C_(0)+.^(2n+1)C_(1)+ . . +.^(2n+1)C_(n)+.^(2n+1)C_(n+1)+ . . .+.^(2n+1)C_(n+1)`
`=(1+1)^(2n+1)=2^(2n+1)`
`implies .^(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)[because.^(n)C_(r)=C_(n-r)]`
`implies1+2xx63+1=2^(2n+1)implies128=2^(2n+1)`
`implies2^(7)=2^(2n+1)implies7=2n+1`
`thereforen=3`