Permutations and Combinations 2 Question 20
20. A student is allowed to select atmost $n$ books from $n$ collection of $(2 n+1)$ books. If the total number of ways in which he can select at least one books is 63 , find the value of $n$.
(1987, 3M)
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Answer:
Correct Answer: 20. $n=3$
Solution:
- Since, student is allowed to select at most $n$ books out of $(2 n+1)$ books.
$\therefore \quad{ }^{2 n+1} C _1+{ }^{2 n+1} C _2+\ldots .+{ }^{2 n+1} C _n=63$
We know ${ }^{2 n+1} C _0+{ }^{2 n+1} C _1+\ldots . .+{ }^{2 n+1} C _{2 n+1}=2^{2 n+1}$
$\Rightarrow 2\left({ }^{2 n+1} C _0+{ }^{2 n+1} C _1+{ }^{2 n+1} C _2+\ldots+{ }^{2 n+1} C _n\right)=2^{2 n+1}$
$\Rightarrow \quad{ }^{2 n+1} C _1+{ }^{2 n+1} C _2+\ldots+{ }^{2 n+1} C _n=\left(2^{2 n}-1\right)$
From Eqs. (i) and (ii), we get
$$ \begin{array}{rlrl} \Rightarrow & & 2^{2 n}-1 & =63 \\ \Rightarrow & 2^{2 n} & =64 \\ \Rightarrow & 2 n & =6 \\ & n & =3 \end{array} $$
