## permutationA particular ordering of a collection of objects. For example, if an athlete has won three medals, a bronze one (B), a silver one (S), and a gold one (G), there are six ways they can be permuted or lined up: BSG, BGS, SBG, SGB, GBS, and GSB. If six people want to sit on the same park bench, there are 720 ways in which they can organize themselves. In general, n
things can be permuted in n × (n - 1) × (n
- 2) × ... × 2 × 1 = n! ways (where "!" is the symbol
for factorial). How about if there are
n distinct objects but we want to permute them in groups of k
(where k
n). How many ways can that be done? The first member of the group
can be picked in n ways because there are n objects to pick
from. The second member can be filled in (n - 1) ways since one of
the n elements has already been taken. The third member can be filled
in (n - 2) ways since 2 elements have already been used, and so.
This pattern continues until there are k things have been chosen.
This means that the last member can be filled in (n - k +
1) ways. Therefore a total of n (n - 1)(n - 2) ...
(n - k + 1) different permutations of k objects, taken
from a pool of n objects, exist. If we denote this number by P(n,
k), we can write P(n, k) = n! / (n -
k)! ## Related entry• combinatorics | |||||

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