Placing eight apples into three baskets

Question

You place 8 identical apples into 3 distinct baskets. A basket may hold any number, including none. In how many ways can the apples be distributed?

Accepted Answer

A distribution is a count list (x1, x2, x3) of non-negative whole numbers with x1 + x2 + x3 = 8. Represent it with 8 stars (apples) and 2 bars (dividers between the 3 baskets). Every arrangement of these 8 + 2 = 10 symbols is one distribution, so choose which 2 are bars: \binom{8 + 3 - 1}{3 - 1} = \binom{10}{2} = \frac{10 	imes 9}{2} = 45. Takeaway. Stars and bars: n identical items into k groups (empties allowed) is \binom{n+k-1}{k-1}. Here \binom{10}{2} = 45.

Placing eight apples into three baskets

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