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 $126$

$N_1={}^5{c}_1\times {}^4{c}_4=5\times 1=5$

$N_2={}^5{c}_2\times {}^4{c}_3=10\times 4=40$

$N_3={}^5{c}_3\times {}^4{c}_2=10\times 6=60$

$N_4={}^5{c}_4\times {}^4{c}_1=5\times 4=20$

$N_5={}^5{c}_5\times {}^4{c}_0=1\times 1=1$

$N_1+N_2+N_3+N_4+N_5=5+40+60+20+1=126$