n-Permutation of n objects: P (n, n) = n! = number of ways to place n nonattacking rooks on an n × n chessboard. r-Permutation of n objects: P (n, r) = n(n − 1)...(n − r + 1) = number of ways to place r nonattacking rooks on an r × n chessboard, r ≤ n. C(n, r)P (n, r) = \frac{[n(n−1)...(n−r+1)]^{2}}{r!} = number of ways to place r nonattacking rooks on an n × n chessboard, r ≤ n. 2.4 Permutations of Multisets Thm 2.4.1: Let A = {∞ · 1, ∞ · 2, ..., ∞ · k} The number of r permutations of A = k^{r}. Thm 2.4.2: Let B = {n_{1} · 1, n_{2} · 2, ..., n_{k} · k} n = n_{1} + n_{2} + ... + n_{k} The number of n-permutations of B = \frac{n!}{n_{1}!n_{2}!···n_{k}!}. If want r-permutations of B, need to use or statements or technique from later chapter. Combinations: order does NOT matter ( \begin{matrix}n \\ r \end{matrix} ) = # of r combinations of \{1, 2, ..., n\}

= subsets of {1, 2, ..., n} containing exactly r elements.

2^{n} = Σ_{i=0}^{n} ( \begin{matrix}n \\ i \end{matrix} ) = # of subsets of \{1, 2, ..., n\}.

Pascal’s Triangle: C(n, r) = C(n − 1, r − 1) + C(n − 1, r) 2.4 Combinations of Multisets Let A = {∞ · 1, ∞ · 2, ..., ∞ · k}

The number of r combinations of A = ( \begin{matrix}r + k − 1 \\ r \end{matrix} )

= # of solutions to x_{1} + x_{2} + ... + x_{k} = r such that x_{i} ≥ 0, x_{i} ∈ Z = # of permutations of {r · 1, (k − 1) · +} = partitions of r indistinguishable objects into k distinguishable boxes.

if need x_{i} ≥ a_{i} replace x_{i} with y_{i} + a_{i} since y_{i} = x_{i} − a_{i} ≥ 0