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

Let three digit number is $x y z$

$x+y+z=10 ; \quad x \geq 1, y \geq 0 \quad z \geq 0$    ...(1)

Let $\mathrm{T}=\mathrm{x}-1 \Rightarrow \mathrm{x}=\mathrm{T}+1$ where $\mathrm{T} \geq 0$

Put in (1)

$\mathrm{T}+\mathrm{y}+\mathrm{z}=9 ; \quad 0 \leq \mathrm{T} \leq 8,0 \leq \mathrm{y}, \mathrm{z} \leq 9$

No. of non negative integral solution

$={ }^{9+3-1} \mathrm{C}_{3-1}-1($ when $\mathrm{T}=9)$

$=55-1=54$