Permutation

Distribute $k$ distinct balls into $n$ distinct cells (each cell can contain at most one ball)

k-permutations of n (חליפה) - $P (n, k) =_{n} P_{k} = \frac{n !}{( n - k )!} = (n)_{k}$
n-permutations of n (תמורה) - $P (n) = n!$

Permutation with repetition

Distribute $k$ distinct balls into $n$ distinct cells (when there is no restriction on the capacities of the boxes, that is, a cell can be empty. ^[If a cell cannot be empty and must contain at least one object, then the number is $n! {n k}$ , which is also the number of onto function] )

$n^{k}$

Restrictions

Distribute $k$ distinct balls into $n$ distinct cells, where the first $a$ cells will together contain at least $b$ balls.

$\sum_{i = 0}^{a - 1} (b + i k) \cdot a^{(b + i)} \cdot (n - a)^{(k - (b + i))}$

Permutations of multisets

$P (n; k_{1}, k_{2}, ..., k_{i}) = (k _{1} , k _{2} , \dots , k _{i} n) = \frac{n !}{k _{1} ! k _{2} ! \dots k _{i} !} = (k _{1} k _{1}) (k _{2} k _{1} + k _{2}) \dots (k _{i} k _{1} + k _{2} + \dots + k _{i})$

See also: multinomial coefficient ^[also called multiset permutation or multinomial theorem or תמורה עם חזרות or המקדם המולטינומי]

$k$ distinct balls into $n$ distinct cells, where the order of balls within the cells is matter $k! (k n + k - 1) = \frac{( n + k - 1 )!}{( n - 1 )!} = (n + k - 1)^{\underline{k}}$

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Permutation

Permutation with repetition

Restrictions

Permutations of multisets

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