How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters? Provide your answer as a sum of binomial coefficients in the form \(\sum\limits_{k=0}^{n}\binom{m}{k}^{p}\), and then calculate the value of m + n + p.

Let's think step by step and output the final answer within \boxed{}.
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