permutation and combination question?

Question

How many ways are there to distribute 40 identical marbles among 4 children without restriction

Answer 1

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There would be (43 C 3) = 43!/(40!*3!) = 12341 ways to distribute the marbles.

This is the same as the number of solutions to the equation w + x + y + z = 40,

where w, x, y and z are all integers greater than or equal to 0. Let each marble

be represented by an o and let the + signs be the dividers between the collection

of marbles each child ends up with. Then every distribution of marbles will be

an arrangement of 40 o's and 3 +'s, and there are (40+3)! / (40!*3!) = (43 C 3)

such arrangements.