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The number of derangements of $n$ items is $!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$.

If there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both.

The number of ways to arrange $n$ distinct objects is $n!$ (factorial).

The expansion of $(x+y)^n$ is given by $\sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k$.

For $n \geq r$, $\binom{r}{r} + \binom{r+1}{r} + ... + \binom{n}{r} = \binom{n+1}{r+1}$.

$n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ for large $n$.

$C_n = \frac{1}{n+1}\binom{2n}{n}$ is the $n^{th}$ Catalan number.

$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ for $1 \leq k \leq n$.

If a set has $n$ objects with $n_1, n_2, ..., n_r$ indistinguishable objects respectively, the number of distinct permutations is $\frac{n!}{n_1! n_2! ... n_r!}$.

$\binom{m+n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k}$ for $0 \leq r \leq m+n$.

The expansion of $(x_1+x_2+...+x_k)^n$ is given by $\sum \frac{n!}{n_1!n_2!...n_k!}x_1^{n_1}x_2^{n_2}...x_k^{n_k}$, where the sum is taken over all non-negative integer sequences $(n_1,...,n_k)$ such that $\sum n_i = n$.

The number of combinations of $n$ items taken $k$ at a time with repetition allowed is $\binom{n+k-1}{k}$.

The number of ways to divide a set of $n$ objects into $k$ non-empty subsets is given by Stirling numbers of the second kind, $S(n, k)$.

The number of ways to choose $k$ items from $n$ items without regard to order is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.