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A set contains (2n + 1) elements. The number of subsets of the set which contain at most n elements is

(A)   2n

(B)   2n+1

(C)   2n−1

(D)   22n

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Correct option  (D) 22n

The number of subsets of the set which contain at most n elements is

2n+1C0 + 2n+1C1 + 2n+1C2 + … + 2n+1Cn = N (say)

We have

2N = 2(2n+1C0 + 2n+1C1 + 2n+1C2 + …+ 2n+1Cn)

= (2n+1C0 + 2n+1C2n+1) + (2n+1C1 + 2n+1C2n) + … + (2n+1Cn + 2n+1Cn+1)

(Since, nCr = nCn–r)

= 2n+1C0 + 2n+1C1 + 2n+1C2 + … + 2n+1C2n+1 = 22n+1

⇒ N = 22n

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