# 2009 UNCO Math Contest II Problems/Problem 1

## Problem

How many positive $3$-digit numbers $abc$ are there such that $a+b=c$ For example, $202$ and $178$ have this property but $245$ and $317$ do not.

## Solution

Here, we need to find $a,b\in \Bbb N_0$ such that $1\le a\le 9$ , $0\le b\le 9$ and $a+b=c$ where $c\in \Bbb N, c\le 9$.

If we place $a=1$, then we can place $0,1,2,3,4,5,6,7,8$ as $b$, i.e. in $9$ ways.

Similarly, if we place $a=2$, we can place $b=0,1,2,3,4,5,6,7$ i.e. in $8$ ways.

$$\dots$$

If, we place $a=9$, we have the only choice $b=0$, in $2$ ways.

So, in order to get the number of possibilities, we have to add the no. of all the possibilities we got, i.e. the answer is $$\color{red}{1+2+3+4+5+6+7+8+9=\frac {9\times 10}{2}}=\color{blue}{45}$$