Question: 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 one book is 63, then the value of $ n $ is [IIT 1987; RPET 1999; Pb. CET 2003; Orissa JEE 2005]
Options:
A) 2
B) 3
C) 4
D) None of these
Show Answer
Answer:
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 the choice to select one, two, three, ……, n books. Thus, if T is the total number of ways of 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_1+{{}^{2n+1}}C_2+…..+{{}^{2n+1}}C _{n}+{{}^{2n+1}}{C _{n+1}} $ $ {{+}^{2n+1}}{C _{n+2}}+….+{{}^{2n+1}}{C _{2n+1}}={{(1+1)}^{2n+1}}={2^{2n+1}} $ or $ ^{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}} $
Þ $ 1+2(T)+1={2^{2n+1}} $
Þ $ 1+T=\frac{{2^{2n+1}}}{2}=2^{2n} $
Þ $ 1+63=2^{2n}\Rightarrow 2^{6}=2^{2n}\Rightarrow n=3 $ .
BETA
