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I have $8$ pieces of strawberry candy (all identical) and $7$ pieces of rhubarb candy (all identical). Find the number of ways I can distribute this candy to $4$ children.

 Nov 4, 2020
 #1
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By stars-and-bars, the number of ways is C(9,4)*C(8,4) = 8820.

 Nov 4, 2020
 #2
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This is my attempt at this question:

 

Any such distribution of the strawberry candy can be encoded in a string such as \[XX|XXX|X|XX\]
of 8 copies of $X$ along with three dividers. (The above example corresponds to giving 2, 3, 1, and 2 candies to the four children, in order.) Since we are allowed to give a child no candy if we wish, the three dividers and the 8 X's can be in any order (in particular we're allowed to have two dividers next to each other). There are then $\binom{11}{3} =165$ ways of arranging 8 X's and 3 |'s, and thus 165 ways of distributing the strawberry candy.

Similarly, distributing the rhubarb candy corresponds to arranging 7 Y's and 3 |'s, so we have $\binom{10}{3} = 120$ ways of distributing the rhubarb candy.

This yields a total of $165\cdot 120 = \boxed{19800}$ ways of distributing the candy.

 

I hope this helped! angel

 Nov 6, 2020

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