Question

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

• A. $280$
• B. $256$
• C. $479$
• D. $685$

Answer 1

The correct option is D $685$

Given $15<x_1+x_2+x_3\le 20$ such that $x,y,z$ are positive integers.

$\Rightarrow x_1+x_2+x_3$ = $16,17,18,19,20$

No. of ways = ${}^{n-1}{C}_r$

Where $n=16$ to $20$ $\xi$ $r=3$

$\Rightarrow$ Total $={}^{15}{C}_2+{}^{16}{C}_2+{}^{17}{C}_2+{}^{18}{C}_2+{}^{19}{C}_2$

$=105+120+136+153+171$

$=685$

Hence, the answer is $685.$