Partial Permutations

combinatorics

Partial Permutations

Given $n$ distinct objects, how many arrangements of length $k$ can be formed? Two arrangements are considered different if they differ in at least one element, or consist of the same elements but in a different order.

Such arrangements are called partial permutations (without repetition) and are denoted $A_n^k$.

To construct one, we need to make $k$ choices:

at the first step we choose from $n$ objects
at the second step – from $n - 1$ objects (repetition is not allowed)
at the $k$-th step – from $n - k + 1$ objects

Thus, $A_n^k = n \cdot (n - 1) \cdot … \cdot (n - k + 1) = \frac{n!}{(n - k)!}$

Problem

A scientific society consists of 25 members. A president, vice-president, scientific secretary, and treasurer must be elected. In how many ways can this be done?

$A_{25}^4 = 25 \cdot 24 \cdot 23 \cdot 22 = 303 \ 600$