Mathematics - Permutation Combination Question with Solution | TestHub

We have,

$x\ge 1, y\ge 3, z\ge 4$

$\Rightarrow x-1\ge 0, y-3\ge 0, z-4\ge 0$

So,

$x+y+z=21$

$\Rightarrow \left(x-1\right)+\left(y-3\right)+\left(z-4\right)=13$

So, number of non-negative integral solutions is

$={}^{13+3-1}C_{3-1}={}^{15}C_2=\frac{15\times 14}{2}=105$