Permutations and Combinations 2 Question 9
9. Let $S={1,2,3, \ldots \ldots, 9}$. 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 $N _1+N _2+N _3+N _4+N _5=\quad$ (2017 Adv.)
(a) 210
(b) 252
(c) 126
(d) 125
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Answer:
Correct Answer: 9. (c)
Solution:
- $N _i={ }^{5} C _k \times{ }^{4} C _{5-k}$
$$ \begin{aligned} & N _1=5 \times 1 \\ & N _2=10 \times 4 \\ & N _3=10 \times 6 \\ & N _4=5 \times 4 \\ & N _5=1 \\ & N _1+N _2+N _3+N _4+N _5=126 \end{aligned} $$
