Ever wondered how huge numbers get even huger? Let’s take a quick tour into factorials and answer what is the factorial of 100 in a way that’s friendly, useful, and a little bit fun.
The phrase what is the factorial of 100 pops up when people want to see how combinatorics, probability, and computing collide. Spoiler: the number is astronomically large, and it shows up in surprising places from cryptography to counting problems.
What Factorial Means and How to Calculate It
Factorial is a simple rule with powerful results. The factorial of a positive integer n, written as n!, means multiply all whole numbers from 1 up to n. So:
1! = 1
2! = 2 × 1 = 2
3! = 3 × 2 × 1 = 6
To get to 100!, you multiply every integer from 1 to 100. That makes 100! = 100 × 99 × 98 × ... × 2 × 1. Doing that by hand is impossible for most people, so computers or calculators handle it.
How big is 100!? In exact form it’s a 158‑digit number:
93326215443944152681699238856266700490715968264381621468592963895217\ 59999322991560894146397615651828625369792082722375825118521091686400000000000000
That entire block is 100! written out precisely and it shows how factorials explode in size.
Why 100 Factorial Appears Everywhere
Counting permutations: 100! is the number of different ways to arrange 100 unique items in a line. That’s useful in scheduling, genetics, and puzzles.
Probability and combinations: Factorials underlie combinations and permutations formulas like nCr and nPr.
Stirling’s approximation: For large n, n! is approximated by n^n e^-n sqrt(2πn); this makes estimates tractable without exact values.
Coding and cryptography: Large factorials test big‑integer libraries and performance of arbitrary-precision arithmetic.
Curiosity and benchmarks: 100! is a common example when teaching big numbers or benchmarking math software.
Quick Rules and Tricks
Use programming languages with big integer support for exact values.
Use Stirling’s approximation for quick estimates and logs.
To count trailing zeros, count factors of 5 in 100: floor(100/5) + floor(100/25) = 20 + 4 = 24 trailing zeros in 100!
Factorials grow faster than exponential functions like 2^n after a point.
Use logs to compare sizes: log10(100!) ≈ 157.97, so 100! has 158 digits.
Did You Know? The number of trailing zeros in n! is determined by how many times the factors 2 and 5 pair up. Since 2s are abundant, counting 5s gives the answer. For 100!, there are 24 trailing zeros.
Mini Q&A
Q: Is 0! defined? A:
Yes. By convention 0! = 1, which makes many formulas consistent.
Q: Can factorials be defined for non-integers?
A: Yes. The Gamma function extends factorials to non-integers: Γ(n+1) = n!.
Q: Why does 100! have so many digits?
A: Multiplying 100 consecutive integers compounds growth multiplicatively, so the number of digits rises quickly.
The History and Cultural Notes
The factorial notation “!” was introduced by mathematician Christian Kramp and popularized by others in the early 19th century.
Factorials underlie classic problems like the “100 prisoners and hats” and the famous “birthday problem,” which are staples in recreational math and teaching.
In computing culture, printing 100! is a common first test of big-number libraries and interview questions about integer overflow.
Factorials appear in statistical formulas such as permutations, combinations, and expansions of power series like e^x.
Personal Reflection
I remember first seeing 100! written out in a textbook and feeling equal parts awe and confusion. It’s one thing to know the rule; it’s another to see a single multiplication rule produce a number longer than most phone numbers. That sense of scale is why I still enjoy teaching factorials—small rules, massive consequences.
Final Thoughts and Takeaway
The factorial of 100, 100!, is a concrete example of how simple mathematical rules create enormous numbers that matter in counting, probability, and computing. Whether you need the exact 158‑digit value or just an approximation, what is the factorial of 100 points to deeper ideas about growth, approximation, and numerical tools.
Curious how factorials show up in your favorite hobby or job? Share a story or question below and let’s explore it together.