Question

The number of negative integral solution of the equation $x_1+x_2+x_3+x_4+15=0$ is

• A. $364$
• B. ${}^{15}{C}_3$
• C. ${}^{15}{C}_4$
• D. $noneofthese$

Answer 1

The correct option is A $364$

Since $x_1+x_2+x_3+x_4=-15$
Let's distribute $-1$ to each of the $4$ variables hence now the equation becomes
$x_1+x_2+x_3+x_4=-11$
Let $-p_1-p_2-p_3-p_4=11$
now since 11 is to be distributed in 4 variables hence no. of ways =$\left(\begin{array}{c}n+r-1\\ r-1\end{array}\right)$

where n=number to be distributed, and r= no. of variables
=$\left(\begin{array}{c}11+4-1\\ 4-1\end{array}\right)=\left(\begin{array}{c}14\\ 3\end{array}\right)=364$
Hence, option 'A' is correct.