Counting Methods

Principle of Multiplication

If an object is to be chosen from a set of m objects and a second object is to be chosen from a different set of n objects, then there are mn ways of choosing both objects simultaneously.

For example, each time a coin is flipped, there are two possible outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, then the experiment has 28 possible outcomes, where each of these outcomes is a list of heads and tails in some order.

Factorial

A symbol that is used with the multiplication principle is the factorial. If nis an integer greater than 1, then n factorial, denoted by the symbol n!, is defined as the product of all the integers from 1 to n.

The factorial is useful for counting the number of ways that a set of objects can be ordered. If a set of n objects is to be ordered from 1st to nth, then there are n choices for the 1st object, (n-1) choices for the 2nd object, (n-2) choices for the 3rd object, and so on, until there is only 1 choice for the nth object. For example, the number of ways of ordering the letters A, B, and C is 3!, or 6. These orderings are called the permutations of the letters A, B, and C.

Permutation

A permutation is a selection process in which objects are selected one by one in a certain order.

permutation

Combination

Permutations means how many different arrangements can be created from a group of items. Combinations  means how many different ways of selecting items from a larger group.

combination