Question

The total number of $3$-digit numbers, whose sum of digits is $10$, is

Answer 1

Let $xyz$ be the $3$ digit number
$x+y+z=10$ where $x\ge 1,y\ge 0,z\ge 0$
Let $x-1=t\Rightarrow x=1+t$
$\Rightarrow t+y+z=9$ where $t\ge 0,y\ge 0,z\ge 0$
Total number of non negative integral solution
$={}^{9+3-1}{C}_{3-1}={}^{11}{C}_2=55$
But for $t=9,x=10$ not possible
$\therefore$ Total numbers = $55-1=54$