Solveeit Logo
Login

Question

Question: A set contains \(2 n + 1\) elements. The number of subsets of this set containing more than n elem...

A set contains 2n+12 n + 1 elements. The number of subsets of this set containing more than n elements is equal to.

A

2n−12 ^ { n - 1 }

B

2n2 ^ { n }

C

2n+12 ^ { n + 1 }

D

22n2 ^ { 2 n }

Answer

22n2 ^ { 2 n }

Explanation

Solution

Let the original set contains (2n+1)( 2 n + 1 ) elements, then subsets of this set containing more than n elements, i.e., subsets containing (n+1)( n + 1 ) elements, (n+2)( n + 2 ) elements, ……. (2n+1)( 2 n + 1 ) elements.

∴ Required number of subsets

=2n+1Cn+2n+1Cn−1+…+2n+1C1+2n+1C0= { } ^ { 2 n + 1 } C _ { n } + { } ^ { 2 n + 1 } C _ { n - 1 } + \ldots + { } ^ { 2 n + 1 } C _ { 1 } + { } ^ { 2 n + 1 } C _ { 0 }

=2n+1C0+2n+1C1+2n+1C2+…+2n+1Cn−1+2n+1Cn= { } ^ { 2 n + 1 } C _ { 0 } + { } ^ { 2 n + 1 } C _ { 1 } + { } ^ { 2 n + 1 } C _ { 2 } + \ldots + { } ^ { 2 n + 1 } C _ { n - 1 } + { } ^ { 2 n + 1 } C _ { n }

=12[(1+1)2n+1]= \frac { 1 } { 2 } \left[ ( 1 + 1 ) ^ { 2 n + 1 } \right] =12[22n+1]=22n= \frac { 1 } { 2 } \left[ 2 ^ { 2 n + 1 } \right] = 2 ^ { 2 n }.