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 at least one book is 63 , find the value of n.

Question

DivineJEE · Accepted Answer

Number of ways of selecting 3 balls with at least one black ball = Total number of selections - Number of selections with no black ball = ^9C_3 - ^6C_3 = 84 - 20 = 64. ^2n + 1C_1 + ^2n + 1C_2 + ... ... + ^2n + 1C_n \ = (1 / 2)lbrack 2 * ^2n + 1C_1 + 2 * ^2n + 1C_2 + ... .. + 2 * ^2n + 1C_n rbrack \ = (1 / 2)lbrack ^2n + 1C_1 + ^2n + 1C_2 + ... ... .. + ^2n + 1C_n . . + ^2n + 1C_1 + ^2n + 1C_2 + ... ... .. + ^2n + 1C_n rbrack ^2n + 1C_1 = ^2n + 1C_2n ^2n + 1C_2 = ^2n + 1C_2n - 1 \_\_\_\_ ^2n + 1C_n = ^2n + 1C_n + 1 \ = (1 / 2)lbrack ^2n + 1C_0 + ^2n + 1C_1 + ... ... + ^2n + 1C_n . \ . + ^2n + 1C_n + 1 + ^2n + 1C_n + ... ... + ^2n + 1C_2n + ^2n + 1C_2n + 1 - 2 rbrack \ = (1 / 2)( 2^2n + 1 - 2 ) = 2^2n - 1 = 63 \ => 2^2n = 64 = 2^6 => 2n = 6 => n = 3

Explanation

For (2 leq r leq n,binom{n}{r} + 2binom{n}{r - 1} + binom{n}{r - 2} =)

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