N factorial

Definition

n factorial, denoted as $n!$, is the product of all positive integers from 1 to $n$. It is commonly used in permutations and combinations to count possible arrangements.

5 Must Know Facts For Your Next Test

1. 0! is defined to be 1.
2. $n!$ grows very rapidly with increasing $n$.
3. The formula for permutations of $n$ items taken $r$ at a time is $\frac{n!}{(n-r)!}$.
4. $n!$ can be used to calculate combinations using the formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.
5. In sequences and series, factorials often appear in formulas for terms involving products.

Review Questions

• What is the value of 5 factorial?
• How do you express the number of ways to arrange 7 books on a shelf using factorial notation?
• What is the relationship between permutations and factorials?

Related terms

Permutation:

An arrangement of objects in a specific order. The number of permutations of $n$ objects taken $r$ at a time is given by $\frac{n!}{(n-r)!}$.

Combination:

A selection of items without regard to order. The number of combinations of $n$ objects taken $r$ at a time is given by $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

Binomial Coefficient:

A coefficient that appears in the binomial theorem, denoted as $\binom{n}{k}$, representing the number of ways to choose $k$ elements from a set of $n$ elements.