Question

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

Answer 1

Let three digit number is xyz
$x+y+z=10 ; x \geq 1, y \geq 0 z \geq 0 \,\,$ .....(1)
Let $ T=x-1$
$ \Rightarrow x=T+1$ where $T \geq 0$
Put in (1)
$T + y + z =9 ; \leq T \leq 8,0 \leq y , z \leq 9$
No. of non negative integral solution
$={ }^{9+3-1} C _{3-1}-1$ ( when $T =9)$
$=55-1=54$