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Question

# The numb er of ways of selecting n things out of 3 n things, of which n are of one kind. and n are of a second kind and the rest unlike,

A
n2n1
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B
(n1)2n1
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C
(n+2)2n1
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D
(n+1)2n1
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Solution

## The correct option is C (n+2)2n−1Suppose k objects have been chosen out of the n dissimilar objects.Then we have to choose n−k objects from out of the two other sets. The choices would go like O from the fist set and (n−k) from other.∴n∑k=0(n−k+1)(nk)=n∑k=1(n+1)(nk)−n∑k=0k(nk)The first sum is just (n+1)2n. The second sumn∑k=0k(nk)=n∑k=0(n−k)(nk)=12n∑k=0n(nk)=n×2n−1Hence the required number of ways are (n+2)2n−1.

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