Question 13
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and so on. We denote these by $\mathrm{TH},$ $\mathrm{HH},$ and so on. For example, in the sequence $\mathrm{HHTTHHHHTHHTTTT}$ of $15$ coin tosses, there are two $\mathrm{HH},$ three $\mathrm{HT},$ four $\mathrm{TH},$ and five $\mathrm{TT}$ subsequences. How many different sequences of $15$ coin tosses contain exactly two $\mathrm{HH},$ three $\mathrm{HT},$ four $\mathrm{TH},$ and five $\mathrm{TT}$ subsequences?