**Introduction to Combinations and Permutations**

In probability, it is often necessary to choose small sets from large ones, or to figure out the number of ways in which certain sets of outcomes can take place. Permutations and combinations are the two most common ways this is done.

**Factorial**

When working with multiple possibilities, it’s necessary to be familiar with a function called the *factorial* . This function applies only to the natural numbers. (It can be extended to more values, but then it is called the *gamma function* .) The factorial of a number is indicated by writing an exclamation point after it.

If *n* is a natural number and *n* ≥ 1, the value of *n* ! is defined as the product of all natural numbers less than or equal to *n* :

*n* ! = 1 × 2 × 3 × 4 × . . . × *n*

If *n* = 0, then by convention, *n* ! = 1. The factorial is not defined for negative numbers.

It’s easy to see that as *n* increases, the value of *n* ! goes up rapidly, and when *n* reaches significant values, the factorial skyrockets. There is a formula for approximating *n* ! when *n* is large:

*n* ! ≈ *n ^{n}* /

*e*

^{n}where *e* is a constant called the *natural logarithm base* , and is equal to approximately 2.71828. The squiggly equals sign emphasizes the fact that the value of *n* ! using this formula is approximate, not exact.

Find practice problems and solutions at Permutations and Combinations Practice Problems - Set 1.

**Permutations vs. Combinations**

**Permutations**

When working with problems in which items are taken from a larger set in specific order, the idea of a permutation is useful. Suppose *q* and *r* are both positive integers. Let *q* represent a set of items or objects taken *r* at a time in a specific order. The possible number of permutations in this situation is symbolized * _{q} P _{r}* and can be calculated as follows:

* _{q} P _{r}* =

*q*!/(

*q*–

*r*)!

**Combinations**

Let *q* represent a set of items or objects taken *r* at a time in no particular order, and where both *q* and *r* are positive integers. The possible number of combinations in this situation is symbolized * _{q} C _{r}* and can be calculated as follows:

* _{q} C _{r}* =

*/*

_{q}P_{r}*r*! =

*q*!/[

*r*!(

*q*–

*r*)!]

Find practice problems and solutions at Permutations and Combinations Practice Problems - Set 2.

Find more practice problems and solutions for these concepts at: Probability Practice Test.

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