Question

Alice 有 $24$ 个苹果。她可以用多少种方式把这些苹果分给 Becky、Chris 和自己，使得三个人每人至少有两个苹果？

Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

$105$

$114$

$190$

$210$

$380$

知识点：隔板法

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先给三个人每人 $2$ 个苹果，剩下 $24 - 3 \cdot 2 = 18$ 个。

设 Alice、Becky 和 Chris 额外得到的苹果数分别为 $a, b, c$ 。那么 $a + b + c = 18,$

这些变量都是非负整数，因为每人至少两个苹果的条件已经满足。

由隔板法，方案数为 $\binom{18 + 3 - 1}{3 - 1} = \binom{20}{2} = 190.$

正确答案是 C。

Let us assign everybody $2$ apples. This leaves us with $24 - 3 \cdot 2 = 18$ apples.

Then, we want to solve $a + b + c = 18,$ where $a, b, c$ are the additional apples given to Alice, Becky, and Chris.

Note that these are all nonnegative, since we already satisfied the only condition we needed to.

We can use stars and bars to get the number of solutions as $\binom{18 + 3 - 1}{3 - 1} = \binom{20}{2} = 190.$

Thus, the correct answer is C.