The total number of 3-digit numbers, whose sum of digits is 10, is: Answer: 54

Question

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

Answer: 54

Accepted Answer

Let the number be $\overline{xyz}$ with $x\ge1$, $x+y+z=10$. Substituting $x=1+t$ ($t\ge0$): $t+y+z=9$, $0\le t,y,z\le9$.

Total non-negative solutions: $\binom{9+2}{2}=\binom{11}{2}=55$.

Subtract the invalid case $t=9$ (giving $x=10$, not a digit): $1$ case.

$$\text{Total} = 55-1 = 54$$

Solutions · 1