Track / Stage 4 / 160 of 340 #420 of 1964

Problem 420

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Multiple choice

5. There are 5 red balls and 5 black balls, numbered $1,2,3,4,5$ respectively, and 4 balls are drawn from them. The probability that the numbers on the drawn balls are all different is:

Pick one

Official solution

5.D.

From 10 balls, choosing 4, there are $\mathrm{C}_{10}^{4}=210$ different ways. If it is required that the numbers of the chosen balls are all different, one can first select 4 numbers from 5 numbers, which has $\mathrm{C}_{5}^{4}$ ways. For each number, there are two colors to choose from, so the number of ways to pick balls with all different numbers is $C_{5}^{4} \cdot 2^{4}=80$ . Therefore, the probability of picking balls with all different numbers is $\frac{80}{210}=\frac{8}{21}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.