Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:
A candy company makes 5 colors of jellybeans, which come in equal proportions. If I grab a random sample of 5 jellybeans, what is the probability that I get exactly 2 distinct colors?

Your answer

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

Solution

Solution:
There are $\binom{5}{2} = 10$ possible pairs of colors. Each pair of colors contributes $2^{5} - 2 = 30$ sequences of beans that use both colors. Thus, the answer is $10 \cdot 30 / 5^{5} = 12 / 125$ .

Answer: $\dfrac{12}{125}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.