Question

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

Answer 1

Compute the required value:

Let three digits are $x,y,z$.

$x+y+z=10$ where $x\ge 1,y\ge 0,z\ge 0$………….$\left(1\right)$

Let $T=x-1$

$\Rightarrow$ $x=T+1$ where $T\ge 0$

Put it in equation $\left(1\right)$,

$T+y+z=9$ where $0\le T\le 8,0\le y,z\le 9$

Thus, total number of $3$ digit numbers$={}^{9+3-1}C_{3-1}-1$ [ subtracting $1$ for $t=9$, $x=10$ not possible ]

$={}^{11}C_2-1$

$=\frac{11\times 10\times 9!}{9!\times 2!}-1$ $\left[{}^{n}C_r=\frac{n!}{(n-r)!\times r!}\right]$

$=54$

Hence total number of $3$ digit numbers, whose sum of digits is $10$ is $54$.