A) 2
B) 3
C) 4
D) 1
Correct Answer: B
Solution :
Since the student is allowed to select at most n books out of\[(2n+1)\]books, \[\therefore \]In order to select one book he has to select one book he has the choice to select one, two, three ... n books, thus if T is the total number of ways selecting one book then \[T{{=}^{2n+1}}{{C}_{1}}{{+}^{2n+1}}{{C}_{2}}+.....{{+}^{2n+1}}{{C}_{n}}=63\]?(i) again the sum of binomial coefficients \[^{2n+1}{{C}_{0}}{{+}^{2n+1}}{{C}_{2}}+.....{{+}^{2n+1}}{{C}_{n}}{{+}^{2n+1}}{{C}_{n+1}}+...\] \[={{(1+1)}^{2n+1}}={{2}^{2n+1}}\] Or \[^{2b+1}{{C}_{0}}+2\left( ^{2n+1}{{C}_{1}}{{+}^{2n+1}}{{C}_{2}}+....{{+}^{2n+1}}{{C}_{n}} \right)\] \[{{+}^{2n+1}}{{C}_{2n+1}}={{2}^{2n+1}}\] \[\Rightarrow \] \[1+2(T0+1={{2}^{2n+1}}\] \[\Rightarrow \] \[1+T=\frac{{{2}^{2n+1}}}{2}={{2}^{2n}}\] \[\Rightarrow \] \[1+63={{2}^{2n}}\] \[\Rightarrow \] \[{{2}^{6}}={{2}^{2n}}\] \[\Rightarrow \] \[n=3\]
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