Question

The number of negative integral solution of equation $x+y+z=-12$ is

• A. $44$
• B. $55$
• C. $66$
• D. None of these

Answer 1

The correct option is B $55$

$x+y+z=-12$

Let $x=x^{\prime }-1,y=y^{\prime }-1$ and $z=z^{\prime }-1$

Substituting above values,

$\Rightarrow$ $(x^{\prime }-1)+(y^{\prime }-1)+(z^{\prime }-1)=-12$

$\Rightarrow$ $x^{\prime }+y^{\prime }+z^{\prime }-3=-12$

$\Rightarrow$ $x^{\prime }+y^{\prime }+z^{\prime }=-9$ ---- ( 1 )

Since, the solution is negative integral solution then, $n=9$

Equation ( 1 ) is in the form of,

$x_1+x_2+x_3+....x_m=n$ is ${}^{n+m-1}{C}_{m-1}$

Here, $m=3,n=9$

$\Rightarrow$ Number of ways $={}^{9+3-1}{C}_{3-1}={}^{11}{C}_2=\frac{11\times 10}{2}=55$

$\therefore$ The number of negative integral solutions are $55$.