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Mathematics Question on permutations and combinations

A student is allowed to select at most nn books from a collection of (2n+1)(2n + 1) books. If the total number of ways in which be can select a book is 255255, then the value of nn equals to

A

66

B

55

C

44

D

33

Answer

44

Explanation

Solution

By the given condition 2n+1C1+ 2n+1C2+..........+ 2n+1C4=255^{2n+1}C_{1}+\,^{2n+1}C_{2}+ ..........+\,^{2n+1}C_{4}=255 Now 2n+1C1= 2n+1C2n [∵ nCr= nCn−r]^{2n+1}C_{1}=\,^{2n+1}C_{2n} \, \left[\because \,^{n}C_{r}=\,^{n}C_{n-r}\right] 2n+1C2= 2n+1C2n−1^{2n+1}C_{2}=\,^{2n+1}C_{2n-1} 2n+1Cn= 2n+1Cn+1^{2n+1}C_{n} =\,^{2n+1}C_{n+1} Adding these, we get 2n+1C0+ 2n+1C1+ 2n+1C2+...........+ 2n+1Cn^{2n+1}C_{0}+\,^{2n+1}C_{1}+\,^{2n+1}C_{2}+...........+\,^{2n+1}C_{n} =2n+1Cn+1+ 2n+1C2n+ 2n+1C2n+1=^{2n+1}C_{n+1}+\,^{2n+1}C_{2n}+\,^{2n+1}C_{2n+1} ⇒2[2n+1C0+ 2n+1C1+..........+ 2n+1Cn]\Rightarrow 2\left[^{2n+1}C_{0}+\,^{2n+1}C_{1}+..........+\,^{2n+1}C_{n}\right] =22n+1=2^{2n+1}