Question

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

Answer 1

Answer: 54

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