Weird limits

Calculus Level 5

\[\large \lim _{ n\rightarrow \infty }{ \binom{n}{7} { p }^{ 7 }{ q }^{ n-7 } }=\frac {{e}^{A}{B}^{C}}{D!}\ \]

Given the above, where \(p+q=1\) and \(np=5\), find \(A+B+C+D\).

Bonus: Generalize for \(r\) and \(m\) in place of 7 and 5 respectively.

Notation: \(\displaystyle \binom{n}{r}=\frac{n!}{(n-r)!r!}\) denotes the binomial coefficient.

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