Five-block tower using exactly two colours

Question

A bin holds $21$ blocks: $7$ red, $7$ blue and $7$ yellow. Five blocks are drawn at random without replacement and stacked into a tower. What is the probability that the tower contains exactly two of the three colours?

Short answer

$\tfrac{5880}{20349}=\tfrac{280}{969}\approx0.289$ . Pick the two colours ( $3$ ways); the number of 5-block selections from those $14$ blocks is $\binom{14}{5}$ , minus the $2\binom75$ that use only one colour. So the probability is $3\big(\binom{14}{5}-2\binom75\big)/\binom{21}{5}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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