1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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

Binomial Theorem

Solution:

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
$2K = 2 \left(^{2n + 1}C_{0 }+^{ 2n + 1}C_{1} +^{ 2n + 1}C_{2} + .... + ^{2n + 1}C_{n}\right)$
$= \left(^{2n + 1}C_{0} + ^{2n + 1}C_{2n + 1}\right) + \left(^{2n + 1}C_{1} +^{ 2n + 1}C_{2n}\right)$
$+ ... + \left(^{2n + 1}C_{n} + ^{2n + 1}C_{n + 1}\right)\quad \left(\because ^{n}C_{r} = ^{n}C_{n -r}\right)$
$= ^{2n + 1}C_{0} + ^{2n + 1}C_{1} + ^{2n + 1}C_{2} + .... + ^{2n + 1}C_{2n + 1}$
$= 2^{2n + 1} \Rightarrow K = 2^{2n}$