Question

If $x_1+x_2+x_3+x_4+x_5=6$, then the difference between the number of non negative integral solutions and the number of positive integral solutions will be

Answer 1

$x_1+x_2+x_3+x_4+x_5=6$
Number of non negative integral solutions
$={}^{6+5-1}{C}_{5-1}={}^{10}{C}_4=210$
Number of positive integral solutions
$={}^{6-1}{C}_{5-1}={}^5{C}_4=5$
Difference $=210-5=205$