Track / Stage 4 / 292 of 340 #552 of 2000

Problem 552

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer

6. From the 11 positive integers $1,2,3, \cdots, 11$ , if 3 different positive integers $a, b, c$ are randomly selected, then the probability that their product $a b c$ is divisible by 4 is $\qquad$ .

Your answer

Official solution

Let $A=\{1,3,5,7,9,11\}, B=\{2,6,10\}, C=\{4,8\}, M=$ “ $a b c$ is divisible by 4”, then $P(\bar{M})=\frac{C_{6}^{3}+C_{6}^{2} \cdot C_{3}^{1}}{C_{11}^{3}}=\frac{65}{165}=\frac{13}{33} \Rightarrow P(M)=\frac{20}{33}$ .

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