a set consists of 2n1 elements if the number of

Question

A set consists of (2n+1) elements. If the number of subsets of this set which consist of at most n elements is 8192. Find out the value of n.

Ans: The following set has (2n + 1) elements. The number of subsets of this set comprising at the most n elements is following by formula

^{2n + 1}C₀ + ^{2n + 1}C₁ + ^{2n + 1}C₂ + ..... ^{2n + 1}Cn = (1/2)2²ⁿ⁺¹ =2 ²ⁿ

This figure is given to be 8192, so 2²ⁿ = 8192 ⇔ 2²ⁿ = 2¹³ ⇔ 2n = 13 ⇔ n = 6.5

 Hence n = 6.5.