HMMT-Feb 2017 comb, Problem 1

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0,1,2, \ldots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

Solution: First, there are $10^{3}=1000$ triples $(a, b, c)$. Now, we should count how many of these triples satisfy $a+b=c$. If $c=0$, we get 1 triple $(0,0,0)$. If $c=1$, we get two triples $(1,0,1)$ and $(0,1,1)$. Continuing, this gives that the total number of triples is $1+2+\cdots+10=55$. Therefore, our final answer is $\frac{55}{1000}=\frac{11}{200}$. $\fbox{\frac{11}{200}}$.

Answer: \frac{11}{200}