Question:
A set contains (2n + 1) elements. If the number of subsets of this set which contain at most n elements is 4096, then the value of n is
A set contains (2n + 1) elements. If the number of subsets of this set which contain at most n elements is 4096, then the value of n is
Updated On: Jul 6, 2022
- 6
- 15
- 21
- None of these
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The Correct Option is A
Solution and Explanation
The number of subsets of the set which contain at most n elements is
$^{2n + 1}C_{0} + ^{2n + 1}C_{1} +^{ 2n + 1}C_{2} + .... + ^{2n + 1}C_{n} = K \left(say\right)$
We have
$2K = 2 \left(^{2n + 1}C_{0 }+^{ 2n + 1}C_{1} +^{ 2n + 1}C_{2} + .... + ^{2n + 1}C_{n}\right)$
$= \left(^{2n + 1}C_{0} + ^{2n + 1}C_{2n + 1}\right) + \left(^{2n + 1}C_{1} +^{ 2n + 1}C_{2n}\right)$
$+ ... + \left(^{2n + 1}C_{n} + ^{2n + 1}C_{n + 1}\right)\quad \left(\because ^{n}C_{r} = ^{n}C_{n -r}\right)$
$= ^{2n + 1}C_{0} + ^{2n + 1}C_{1} + ^{2n + 1}C_{2} + .... + ^{2n + 1}C_{2n + 1}$
$= 2^{2n + 1} \Rightarrow K = 2^{2n}$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.