SATHEE: Permutations and Combinations 2 Question 20

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.

$∴^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots . +^{2 n + 1} C_{n} = 63$

We know $^{2 n + 1} C_{0} +^{2 n + 1} C_{1} + \dots . . +^{2 n + 1} C_{2 n + 1} = 2^{2 n + 1}$

$\Rightarrow 2 (^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n}) = 2^{2 n + 1}$

$\Rightarrow^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = (2^{2 n} - 1)$

From Eqs. (i) and (ii), we get

$\begin{aligned} \Rightarrow & 2^{2 n} - 1 & = 63 \\ \Rightarrow & 2^{2 n} & = 64 \\ \Rightarrow & 2 n & = 6 \\ n & = 3 \end{aligned}$

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Permutations and Combinations 2 Question 20

20. A student is allowed to select atmost n books from n collection of (2n+1) books. If the total number of ways in which he can select at least one books is 63 , find the value of n.

Answer:

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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 $n$ .