Factorials: Growth of n!

What is a factorial?

For a non-negative integer $n$, the factorial is $n! = n \cdot (n - 1) \cdot (n - 2) \cdots 2 \cdot 1$

with the convention $0! = 1$ (the empty product).

Combinatorially, $n!$ counts the number of ways to arrange $n$ distinct objects in a sequence — the number of permutations of $n$ items.

Step-by-step: compute 7!

We multiply in sequence, showing the running product at each step.

Step 1: $7 \cdot 6 = 42$ Step 2: $42 \cdot 5 = 210$ Step 3: $210 \cdot 4 = 840$ Step 4: $840 \cdot 3 = 2520$ Step 5: $2520 \cdot 2 = 5040$ Step 6: $5040 \cdot 1 = 5040$

$\boxed{7! = 5040}$

How fast does $n!$ grow?

Factorial grows super-exponentially — faster than any fixed $a^n$ for constant $a$.

• $1! = 1$
• $2! = 2$
• $3! = 6$
• $4! = 24$
• $5! = 120$
• $6! = 720$
• $7! = 5{,}040$
• $8! = 40{,}320$
• $9! = 362{,}880$
• $10! = 3{,}628{,}800$

Each step multiplies by a larger number, so the growth accelerates. By $n = 20$, $n!$ is already above $2 \times 10^{18}$.

Stirling's approximation

For large $n$, a very accurate estimate is $n! \approx \sqrt{2 \pi n} \, \left(\dfrac{n}{e}\right)^n$

It turns the factorial into something comparable with exponentials. For $n = 7$, Stirling gives $\approx 4980$, within 1.2% of the true $5040$.

Where factorials appear

• Permutations: $P(n, r) = n!/(n-r)!$.
• Combinations: $C(n, r) = n! / (r!(n-r)!)$.
• Taylor series: $e^x = \sum_{k=0}^{\infty} x^k / k!$.
• Probability distributions: Poisson's $\lambda^k e^{-\lambda}/k!$, binomial's $\binom{n}{k}$.

Common mistakes

• Claiming $0! = 0$. The convention is $0! = 1$; the empty product is 1, and it makes formulas like $\binom{n}{0} = 1$ work.
• Computing $n!$ by addition. Factorial is a product, not a sum. $5!$ is $120$, not $1+2+3+4+5 = 15$.
• Overflow. $n!$ explodes fast; $20!$ already exceeds a 64-bit integer when multiplied by much more. Use arbitrary-precision arithmetic or logarithms for large $n$.

Try it in the visualization

Adjust the $n$ slider and watch the product assemble step by step, with a bar graph scaling up (sometimes logarithmically, with the toggle on) to visualize the superlinear climb.