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The number of ways of choosing \mathrm{n} objects out of \mathrm{(3 n+1)} objects of which \mathrm{n}are identical and \mathrm{(2 n+ 1)} are distinct, is

Option: 1

\mathrm{2^{2 n}}


Option: 2

\mathrm{2^{2 n+1}}


Option: 3

\mathrm{2^{2 n+1}}


Option: 4

none of these


Answers (1)

best_answer

If we choose \mathrm{k(0 \leq k \leq n)} identical objects, then we must choose \mathrm{(n-k)} distinct objects. This can be done in \mathrm{{ }^{2 n+1} C_{n-k}} ways. Thus, the required number of ways

\mathrm{=\sum_{k=0}^{n}{ }^{2 n+1} C_{n-k}={ }^{2 n+1} C_{n}+{ }^{2 n+1} C_{n-1}+\ldots+{ }^{2 n+1} C_{0}=2^{2 n}}.

Posted by

HARSH KANKARIA

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