Question

The total number of positive integral solution of $15<x_1+x_2+x_3\le 20$ is equal to

• A. $685$
• B. $785$
• C. $1125$
• D. none of these

Answer 1

The correct option is A $685$

As $15<x_1+x_2+x_3\le 20$
$x_1+x_2+x_3=16+r,r=0,1,2,3,4;$
Now the number of positive integral solution of above equation is:
$={}^{13+r+3-1}{C}_{13+r}={}^{15+r}{C}_{13+r}={}^{15+r}{C}_2$
Thus the total number of solution is :
$\sum _{r=0}^4{}^{15+r}{C}_2={}^{15}{C}_2+{}^{16}{C}_2+{}^{17}{C}_2+{}^{18}{C}_2+{}^{19}{C}_2$
$=\frac{1}{2}\left(\right(15\times 14)+(16\times 15)+(17\times 16)+(18\times 17)+(19\times 18\left)\right)$
$=685$