Question

The number of integral solutions of $x+y+z=0$ with $x\ge -5,y\ge -5,z\ge -5$ is

• A. $134$
• B. $136$
• C. $138$
• D. $140$

Answer 1

The correct option is B $136$

Let $x=p-5,y=q-5$ and $z=r-5$, where $p,q,r\ge 0.$
Then the given equation reduces to $p+q+r=15\cdots \left(1\right)$
Now, we have to find non-negative integral solution of Eq. $\left(1\right)$.
The total number of such solutions is ${}^{15+3-1}{C}_{3-1}={}^{17}{C}_2=136.$