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Counting Rules

When selecting elements of a set, the number of possible outcomes depends on the conditions under which the selection has taken place. There are at least 4 rules to count the number of possible outcomes:

Multiplicative rule

Suppose you have j sets of elements, n1 in the first set, n2 in the second set, ... and nj in the jth set. Suppose you wish to form a sample of j elements by taking one element from each of the j sets. The number of possible sets is then defined by

n1n2* ...nj.


Permutation rule

The arrangement of elements in a distinct order is called permutation.

Given a single set of n distinctively different elements, you wish to select k elements from the n and arrange them within k positions. The number of different permutations of the n elements taken k at a time is denoted Pkn and is equal to


Partitions rule

Suppose a single set of n distinctively different elements exists. You wish to partition them into j sets, with the first set containing n1 elements, the second containing n2 elements, ..., and the jth set containing nj elements. The number of different partitions is


where n1 + n2 + ...+ nj =n.

The numerator gives the permutations of the n elements. The terms in the denominator remove the duplicates due to the same assignments in the k sets (multinomial coefficients).

Combinations rule

A sample of k elements is to be chosen from a set of n elements. The number of different samples of k samples that can be selected from n is equal to

The combination rule is a special application of the partition rule, with j=2 and n1=k. From n=n1+n2 it follows that n2 can be replaced by (n-n1). Usually the two groups refer to the two different groups of selected and non-selected samples. The order in which the n1 elements are drawn is not important, therefore there are fewer combinations than permutations (binomial theorem).

Note: The factorial n! is defined by  n! = 1 2 3 ...(n-2) (n-1) n. (0! is defined as 1).

Last Update: 2006-Jšn-17