Possible Duplicate:
Division of Factorials
For a set of $n$ objects of which $n_1$ are alike and one of a kind, $n_2$ are alike and one of a kind, ... , $n_k$ are alike and one of a kind, such that $n_1+n_2+...+n_k=n$, the number of distinguishable permutations is: $$\frac{n!}{n_1!\cdot n_2!\cdot...\cdot n_k!}$$ How come this is always an integer?
$${(n_1 + \dots + n_k)! \over n_1! \cdots n_k!} = \frac{n_1!}{n_1!} \cdot \frac{(n_1 + n_2)!}{n_1!n_2!} \cdot \frac{(n_1 + n_2 + n_3)!}{(n_1 + n_2)!n_3!} \cdots \frac{(n_1 + \dots + n_k)!}{(n_1 + \dots + n_{k-1})!n_k!} =\\ \binom{n_1}{n_1} \cdot \binom{n_1+n_2}{n_2} \cdot \binom{n_1+n_2+n_3}{n_3} \cdots \binom{n_1 + \dots + n_k}{n_k}$$
It follows from the Fundamental Theorem of Arithmetic and the power of a prime in a factorial:
If $p$ is a prime, the power of $p$ at the top is
$$\sum_{m=1}^\infty \lfloor \frac{n_1+n_2+..+n_k}{p^m} \rfloor$$
while the power of $p$ in the denominator is
$$\sum_{m=1}^\infty \lfloor \frac{n_1}{p^m} \rfloor+ \lfloor \frac{n_2}{p^m} \rfloor+...+\lfloor \frac{n_k}{p^m} \rfloor$$
Now, $\lfloor \frac{n_1}{p^m} \rfloor+ \lfloor \frac{n_2}{p^m} \rfloor+...+\lfloor \frac{n_k}{p^m} \rfloor$ is an integer and
$$\lfloor \frac{n_1}{p^m} \rfloor+ \lfloor \frac{n_2}{p^m} \rfloor+...+\lfloor \frac{n_k}{p^m} \rfloor \leq \frac{n_1+n_2+..+n_k}{p^m}$$
thus
$$\lfloor \frac{n_1}{p^m} \rfloor+ \lfloor \frac{n_2}{p^m} \rfloor+...+\lfloor \frac{n_k}{p^m} \rfloor \leq \lfloor \frac{n_1+n_2+..+n_k}{p^m} \rfloor$$
and hence
$$\sum_{m=1}^\infty \lfloor \frac{n_1}{p^m} \rfloor+ \lfloor \frac{n_2}{p^m} \rfloor+...+\lfloor \frac{n_k}{p^m} \rfloor \leq \sum_{m=1}^\infty \lfloor \frac{n_1+n_2+..+n_k}{p^m} \rfloor$$
Now, since all primes appear at a larger power in the numerator, the FTA guarantees that this number is an integer.
You can show that $S_{n_1} \times S_{n_2} \times \ldots \times S_{n_k}$ is a subgroup of $S_n$, the statement then follows from Lagrange's theorem.
