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Problem 388

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

1. A non-transparent box contains 20 small balls of the same size and shape, marked with numbers $1,2, \cdots, 20$ , and two balls are randomly drawn from it. The probability that the sum of the numbers on these two balls is divisible by 3 is ( ).
(A) $\frac{3}{38}$
(B) $\frac{32}{95}$
(C) $\frac{11}{95}$
(D) $\frac{37}{190}$

Your answer

Official solution

- 1. B.

It is easy to know that among the numbers from $1 \sim 20$ , there are 6 numbers divisible by 3, 7 numbers that leave a remainder of 1 when divided by 3, and 7 numbers that leave a remainder of 2 when divided by 3. Therefore, the probability that the sum of two numbers is divisible by 3 is
$P=\frac{\mathrm{C}_{6}^{2}+\mathrm{C}_{7}^{1} \mathrm{C}_{7}^{1}}{\mathrm{C}_{20}^{2}}=\frac{64}{190}=\frac{32}{95} .$

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