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# 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 at least one book is 63. Find the value of n

Dear Student,
Here is the solution:

Since the student is allowed to select at the most n-books out of (2n + 1) books, therefore, he can choose, one book, two books or at the most n books. The number of ways of selecting at least one books are

2n+1C1 + 2n+1C2 + ......... 2n+1Cn = 63 = S (Say)

Again, we know that

2n+1C0 + 2n+1C1 + ......... 2n+1Cn + 2n+1C2n+1 = 22n+1

Now  2n+1C0 = 2n+1C2n+1 = 1

2n+1C1 = 2n+1C2n etc.........

Hence, we have

1 + 1 + 2S = 22n+1

or     2 + 2 x 63 = 22n+1

or     128 = 22n+1 or 27 = 22n+1

=>     2n+1 = 7

or     2n = 6

n = 3

Regards
Team Topperlearning.
Answered by Expert 8th February 2011, 10:24 AM
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