Some Triple Sum

Calculus Level 5

\[ \large \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} \frac{i! j! k!}{(i+j+k+2)!} \]

If the value of the summation above is equal to \(\dfrac{\pi^A}{B} \) for integers \(A\) and \(B\), find the value of \(B - A\).

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