In statistics, the exclamation point (!) is the factorial symbol. When you see a number followed by an exclamation point, like 5!, it means you multiply that number by every positive whole number below it: 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials show up constantly in statistics because they’re the mathematical engine behind counting arrangements, calculating probabilities, and working with many of the field’s most important distributions.
How Factorials Work
The notation is straightforward: n! means “multiply all positive integers from 1 up to n together.” So 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5,040. The values grow extraordinarily fast. While 5! is just 120, 10! is already 3,628,800, and 20! exceeds 2.4 quintillion.
One definition catches most people off guard: 0! equals 1, not 0. This isn’t a typo. It’s a deliberate convention that keeps formulas throughout statistics and probability working correctly. Think of it this way: there is exactly one way to arrange zero objects, which is to do nothing at all.
Counting Arrangements and Combinations
The most common reason you’ll encounter factorials in a statistics course is counting. Whenever you need to figure out how many ways things can be ordered or grouped, factorials do the heavy lifting.
Permutations answer the question “how many ways can I arrange r items out of n, where order matters?” The formula is n! / (n−r)!. If you’re picking 3 winners (first, second, third place) from 10 contestants, that’s 10! / 7! = 720 possible outcomes.
Combinations answer “how many ways can I choose r items out of n, where order doesn’t matter?” The formula is n! / (r!(n−r)!). Choosing 4 people from a group of 10 for a committee (where no one has a special role) gives you 10! / (4! × 6!) = 210 possible groups. This expression, often written as “N choose r,” is called a binomial coefficient, and it appears throughout statistics.
Factorials in Probability Distributions
Several of the probability distributions you’ll encounter in statistics have factorials baked into their formulas. Understanding what the exclamation point means makes these formulas far less intimidating.
The Binomial Distribution
The binomial distribution calculates the probability of getting exactly a certain number of successes across a fixed number of independent trials, like flipping a coin 10 times and wanting exactly 4 heads. The formula uses the combination expression N! / (r!(N−r)!) to count all the different ways those successes could be distributed among the trials. For the coin example, there are 10! / (4! × 6!) = 210 different sequences that produce exactly 4 heads in 10 flips. That count then gets multiplied by the probability of any single such sequence occurring.
The Poisson Distribution
The Poisson distribution models the probability of a given number of events happening in a fixed interval, like how many customers arrive at a store per hour. Its formula places k! (the factorial of the number of occurrences) in the denominator. This factorial term accounts for the number of ways those events could be ordered, preventing the formula from overcounting.
Why Factorials Grow So Fast
Factorials get unmanageably large very quickly. While a calculator handles 10! without trouble, something like 100! is a number with 158 digits. In real statistical work involving large sample sizes, computing exact factorials becomes impractical.
Statisticians get around this using Stirling’s approximation, a shortcut that estimates large factorials without computing them directly. It’s accurate enough for practical purposes and is especially useful when working with logarithms of factorials, which come up in entropy calculations and likelihood functions. Most statistical software handles this behind the scenes, so you rarely need to apply it by hand, but it explains why your computer can breeze through calculations that would otherwise be impossible.
The Gamma Function: Factorials Beyond Whole Numbers
Standard factorials only work for non-negative whole numbers. You can compute 5! or 12!, but what about 3.5! or the factorial of a negative number? In advanced statistics, these situations actually arise.
The gamma function, developed by Euler in 1768, extends factorials to nearly all real and complex numbers (everything except zero and negative integers). For any positive whole number n, the gamma function of n equals (n−1)!, so it perfectly matches regular factorials while also filling in the gaps between them. You’re unlikely to need this in an introductory statistics course, but if you encounter it in more advanced probability work, just know it’s the factorial’s more flexible cousin.
Subfactorials: The Exclamation Point Before a Number
In rare cases, you might see the exclamation point placed before a number, like !n, rather than after it. This notation represents a subfactorial, which counts “derangements,” or the number of ways to arrange objects so that none of them ends up in its original position. For instance, !3 = 2, because if you have three items in positions 1, 2, and 3, there are exactly two ways to rearrange them so every item moves to a new spot. Subfactorials appear occasionally in probability problems involving random matching or shuffling, but they’re far less common than standard factorials.