末尾零

每出现一个末尾 0，就对应一个因子 10，也就是一对 2 和 5。由于 $100!$ 的素因子分解中 2 的个数远多于 5，所以只需数 5 的个数。

在 1 到 100 中，有 20 个数能被 5 整除；其中又有 4 个数能被 $25$ 整除，因此会额外再贡献一个 5。于是 5 的总个数为 $20+4=24$，所以 $100!$ 末尾共有 $\boxed{24}$ 个 0。

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Original Explanation

A trailing zero occurs with each factor of 10, and so we know that every pair of 2 and 5 will give a trailing zero. Intuitively, the frequency of 2 in the prime factorization of 100! is far greater than the frequency of 5, and thus the number of 5s limits and determines the number of trailing zeros.

Among the numbers 1, 2, ..., 100, there are 20 numbers that are divisible by 5 (5, 10, ..., 100). Among these numbers, 4 are divisible by 5 twice (25, 50, 75, 100). Therefore, the frequency of 5 is 24, and thus there are $\boxed{24}$ trailing zeros in 100!.