Question

The number of negative integral solutions of the equation $x_1+x_2+x_3+x_4+x_5+10=0$ is

• A. ${}^{10}{C}_4$
• B. ${}^{10}{C}_5$
• C. ${}^9{C}_5$
• D. ${}^9{C}_6$

Answer 1

The correct option is C ${}^9{C}_5$

Since we are considering negative integers, we can consider
$-(x_1+x_2+x_3+x_4+x_5)=-10$
$x_1+x_2+x_3+x_4+x_5=10$
Where $x_i$ is a positive integer.
Hence the required number of integral non-zero solutions will be
$={}^{10-1}{C}_{5-1}={}^9{C}_4={}^9{C}_5$.