忘记邻居

Forgetting Neighbor

专题: Discrete Math / 离散数学
难度: L3
来源: OpenQuant

题目详情

你的邻居有 $3$ 孩子，但你总是忘记他们多大了。不过你知道他们的年龄都不一样，最大的最多 $15$ 岁，最小的至少 $7$ 岁。他们的年龄有多少种可能？

Your neighbor has $3$ children and you always forget how old they are. However, you know their ages are all different, and the oldest is at most $15$ years old while the youngest is at least $7$ . How many possibilities are there for their ages?

解析

三个孩子的年龄是 $\{7,8,\ldots,15\}$ 中互不相同的 3 个整数。该集合共有 9 个可能年龄，因此可能性总数为

\binom{9}{3}=\boxed{84}.

Original Explanation

The ages are $3$ distinct integers in $\{7,8,\ldots,15\}$ . There are $9$ possible ages, so the number of ways to choose $3$ distinct ages is

\binom{9}{3} = 84.