Let S={1,2,3, ldots ., 9} . For k=1,2, ldots ldots, 5 , let N_K be number of subsets of S , each con

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6.Permutation and Combination

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Let $S=\{1,2,3, \ldots ., 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_K$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

A

$210$

B

$252$

C

$125$

D

$126$

(IIT-2017)

Solution

There are only $4$ even numbers in $S$

$\therefore \quad$ Any subset of 5 elements of $S$ will have at least $1$ odd number.

$\Rightarrow \quad \mathrm{N}_1+\mathrm{N}_2+\mathrm{N}_3+\mathrm{N}_4+\mathrm{N}_5={ }^9 \mathrm{C}_5=126$

Standard 11

Mathematics

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