Question

Number of positive integral solutions of $15<x_1+x_2+x_3\le 20$ is

Answer 1

$15<x_1+x_2+x_3\le 20$
$\Rightarrow x_1+x_2+x_3=16+r$
$r=0,1,2,3,4$
Now number of positive integral solution of
$\Rightarrow x_1+x_2+x_3=16+r$ is
${}^{16+r-1}{C}_{3-1}={}^{15+r}{C}_2$

Thus required number of solutions $=\sum _{r=0}^4{}^{15+r}{C}_2={}^{15}{C}_2+{}^{16}{C}_2+{}^{17}{C}_2+{}^{18}{C}_2+{}^{19}{C}_2=685$