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 $685$

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 $685.$