Factorial

Definition

Factorial, denoted as $n!$, is the product of all positive integers less than or equal to $n$. It is used in permutations, combinations, and other areas of counting theory.

5 Must Know Facts For Your Next Test

1. 0! is defined to be 1.
2. The factorial of a number $n$ (denoted as $n!$) is calculated as $n \times (n-1) \times (n-2) \times ... \times 1$.
3. $n!$ grows very quickly with increasing values of $n$.
4. Factorials are commonly used in calculating permutations and combinations.
5. Stirling's approximation provides an estimate for large factorials: $n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$.

Review Questions

• What is the value of $5!$?
• How is the factorial of 0 defined?
• In what mathematical contexts are factorials frequently used?

Related terms

Permutation: An arrangement of objects in a specific order, often calculated using factorials.

Combination: A selection of items from a larger pool where order does not matter, often calculated using binomial coefficients involving factorials.

Binomial Coefficient: A coefficient giving the number of ways to choose k elements from a set of n elements, denoted as ${n \choose k}$ and calculated using factorials: ${n \choose k} = \frac{n!}{k!(n-k)!}$.