Number of ways a sequence can be arranged to meet the required condition

Question

I toss a coin 15 times and I obtain a strings of Heads(represented by H) and Tails(represented by T). I want to find the number of sequences which can be found such that there are 2 HH, 3 HT , 4 TH and 4 TT subsequences. How do I proceed?

One example of such a sequence can be TTTHHTHTTTHHTTH

Answer 1

Any sequence with 3 $HT$s and 2 $TH$s must have the form $$H^aT^bH^cT^dH^e$$ where $a,\ldots,e$ are positive integers, and $H^a$ represents a sequence of $a$ consecutive heads etc. You need to find all possible triples $(a,c,e)$ yielding 5 $HH$s and all possible pairs $(b,d)$ yielding 4 $TT$s.