Anagrams with two Bs ahead of every O

Question

Consider the distinct rearrangements of the letters of BOOLAHUBBOO. How many of them place at least two of the B's earlier than the first O?

Short answer

$39600$ . Choose the $7$ positions holding B's and O's, $\binom{11}{7}=330$ ways, and fill the other four with L, A, H, U in $4!$ ways. Within the seven positions the pattern must begin BB, after which the last B can go in any of $5$ places among the four O's. Total $330\cdot24\cdot5=39600$ .

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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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