Track / Stage 5 / 144 of 400 #744 of 1964

Problem 744

AIME late

Combinatorics Difficulty 5.3 Find the answer

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?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Answer: $\square$
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 $\frac{55}{1000}=\frac{11}{200}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.