Track / Stage 3 / 11 of 260 #11 of 1964

Problem 11

AMC 10/12, early questions

Combinatorics Difficulty 3.0 Multiple choice

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is \$20$ or more?
$

Pick one

Official solution

The only way to get a total of \$20 or more is if you pick a twenty and another bill, or if you pick both tens. There are a total of $\dbinom{8}{2}=\dfrac{8\times7}{2\times1}=28$ $ways to choose$ 2 $bills out of$ 8 $. There are$ 12 ways to choose a twenty and some other non-twenty bill. There is 1 way to choose both twenties, and also 1 way to choose both tens. Adding these up, we find that there are a total of 14 $ways to attain a sum of$ 20 $or greater, so there is a total probability of$ $\dfrac{14}{28}=\boxed{\textbf{(D) }\frac{1}{2}}\$.$

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