This week I solved a difficult math problem that I've been working on for a while.
How many sequences of 15 coin flips have 5 HH 4TH 3HT 2TT for example HHTTTTH has 1 HH 1 TH 1HT and 3 TT.
Solution: Imagine we only cared about the 4TH and 3HT, the only sequence of flips that works is THTHTHTH. To build up to the actual problem we need to add the 5HH and 2TT, there are four different places we can put the HH, they are where the H's are. So it's equivalent to putting 5 balls into four buckets which is 5 choose four or 56 possibilities for H, using a similar argument we can show that there are 10 possibilites for T. And 56*10=560 which is our answer.
(this isn't explained super well, here's the official solution.)
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