Suppose you have $n$ bins, and $t \in \{0,1,\cdots,n\}$ balls. How many ways are there to arrange the balls, such that we have exactly $j$ streaks of 3.
Examples: $n=5$, $t=4$
OOO_O - includes 1 run of 3.
_OOOO - includes 2 runs of 3.
Combinatorics - Number of streaks
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$\begingroup$
probability
combinatorics
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2So, you're allowed to put at most one ball in each bin? – 2017-02-04
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1This can be done with the Goulden-Jackson cluster method; see for example https://arxiv.org/abs/0810.5113. – 2017-02-04
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0Are you still here, knrumsey? – 2017-02-05
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0Sorry guys. Yes, at most one ball in each bin. Apologies, I should have been more clear. @Tad, thanks I'll check it out. – 2017-02-06