Here's the problem statement
In how many ways can we arrange $5$ $A$s, $7$ $B$s and $4$ $C$s into a $16$-letter word such that there are atleast three $CA$ pairs occurring in a word
The solution mentioned proceeds to calculate the number of words containing exactly four $CA$ pairs and then the number of words containing exactly three $CA$ pairs (which it does by calculating number of ways of inserting $2$ $A$s in $11$ out of $12$ gaps (blank group allowed) of a string containing $3$ $CA$ pairs, $7$ $B$s and one $C$. I agree with this solution, however I cannot figure out why my method fails.
I think calculating number of words that contain $3$ $CA$s pairs and their permutations should suffice, because their permutations do include the cases where another $CA$ pair is formed (so that accounts for $4$ $CA$ pairs formed) and when $C$ does not precedes any of the $A$s, it accounts for the words containing $3$ $CA$ pairs. So, number of words should be $$\frac{13!}{7! \times 3! \times 2!} =102960$$ whereas, according to solution 1 $$\frac{12!}{7! \times 4!} + \frac{11!}{7! \times 3!} \times {12\choose10} = 3960 + 87120 = 91080$$ can someone point out my mistake