Handing out ten candies among four kids

Question

You give out 10 identical candies to 4 children. Any child may receive any number, including zero, and the candies are indistinguishable, so only how many each child gets matters. In how many ways can the candies be distributed?

Accepted Answer

Because the candies are identical, a distribution is just a list of counts (x1, x2, x3, x4) of non-negative whole numbers with x1 + x2 + x3 + x4 = 10. Picture the 10 candies as stars in a row and use 3 dividers ("bars") to split them into 4 groups. For example, \star\star\star\star\star{}\star\star\star\star\star means the kids get 2, 3, 0, 5. Every arrangement of 10 stars and 3 bars gives exactly one valid distribution. So we arrange 10 + 3 = 13 symbols and choose which 3 are bars: \binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = \frac{13 	imes 12 	imes 11}{6} = 286. Takeaway. Placing n identic…

Handing out ten candies among four kids

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