Question

Let $N$ be the number of non-negative integral solution of the equation $x+y+z+\omega =15$ where $x\ge 0,y>5,z\ge 2,$ and $\omega \ge 1$. The unit digit of $N$ is

Answer 1

x + y + z + $\omega$ = 15

$x\ge 0,y\ge 6,z\ge 2,\omega \ge 1$

$x+y^{\prime }+z^{\prime }+{\omega }^{\prime }=6[y^{\prime }\ge 0,z^{\prime }\ge 0,{\omega }^{\prime }\ge 0]$

Number of integral solutions are ${}^9{C}_3=84$

Unit digit of 84 is 4.