Presents for all!

There are 44 children awaiting their presents. 77 elves come one-by-one and each of them gives 11 present to exactly 11 child. Every elf chooses the child randomly.

The probability that each child receives at least one present is of the form AB\dfrac{A}{B} where AA and BB are co-prime positive integers. Find the value of BA.B-A.

Assumptions\textbf{Assumptions}

  • Each elf has a different present.
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