The number of points ( $a, b, c$ ) in space, whose each coordinate is a negative integer such that $

Question

The number of points ( $a, b, c$ ) in space, whose each coordinate is a negative integer such that $a+b+c+12=0$ is

Professor PC Thomas Classes · Answer

Let $a=-x, b=-y, c=-z$. $x+y+z=12$ with $x,y,z \geq 1$. Number of positive integer solutions is $\binom{12-1}{3-1} = \binom{11}{2} = 55$. [FLAG: Check options]