Question

Let $S=\{1,2,3,...,9\}.$ For $k=1,2,...,5,$ 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. $126$

Answer 1

The correct option is D $126$

$S=\{1,2,3,...,9\}$
There are $4$ even and $5$ odd numbers.
$\therefore N_1+N_2+N_3+N_4+N_5={}^5{C}_1\times {}^4{C}_4+{}^5{C}_2\times {}^4{C}_3+{}^5{C}_3\times {}^4{C}_2+{}^5{C}_4\times {}^4{C}_1+{}^5{C}_5\times {}^4{C}_0=126$