Factorial Calculator — n!, Permutations & Combinations

What Is a Factorial Calculator?

A factorial calculator works with n! (read "n factorial") — the product of every positive whole number up to n. This free tool handles four jobs: factorial n! (5! = 120), factorial expressions like n!/(n−k)! (6!/(6−3)! = 120), permutations nPr for ordered arrangements (10P3 = 720), and combinations nCr for unordered selections (10C3 = 120). Every answer comes with the exact steps so you can verify it by hand.

How to Use the Factorial Calculator

The tool has four modes so you can start with whatever you need to find.

1. Choose a mode that matches your problem (or tap a preset to load a worked example).
2. Enter whole numbers — n from 0 to 170, and r (or k) no bigger than n.
3. Select Calculate to generate the result cards.
4. Read the note under the cards for the step-by-step working and a check you can verify by hand.

The built-in check re-derives the answer a second way — n! = n × (n−1)! for factorials, and nCr = nPr / r! for combinations — so a wrong keystroke shows up immediately.

What Is a Factorial?

A factorial counts arrangements. The factorial of n, written n!, is the product of every positive whole number up to n: n! = n × (n−1) × … × 2 × 1. It answers the question "in how many orders can n different things be arranged?" — 5 books on a shelf can be ordered in 5! = 120 ways, because there are 5 choices for the first spot, 4 for the second, 3 for the third, 2 for the fourth, and 1 for the last.

Factorials grow shockingly fast. 10! is already 3,628,800, 20! is about 2.43 × 10¹⁸, and 52! is about 8.066 × 10⁶⁷ — a 68-digit number. The table below shows the first eleven values exactly.

Why 0! = 1 (and Why It Is Not 0)

Zero factorial looks like it should be 0 — but it is 1, by definition, and the definition is not arbitrary. First, 0! is the empty product: the product of no numbers at all, and the empty product is 1, just as the empty sum is 0. Second, the recurrence n! = n × (n−1)! forces it: 1! = 1 × 0! only works if 0! = 1. Third, it keeps the counting formulas working: there is exactly 1 way to choose 0 items from n (choose nothing), so nC0 = n!/(0! × n!) = 1 only if 0! = 1. The calculator's 0! preset shows this directly.

How Big Is 52! Really?

52! ≈ 8.066 × 10⁶⁷ — a 68-digit number. It is often compared with the roughly 10⁸⁰ atoms in the observable universe, but get the direction right: 52! is smaller than the atom count (52!/10⁸⁰ ≈ 8 × 10⁻¹³). The honest shock is time: there have been only about 4.35 × 10¹⁷ seconds since the Big Bang, so 52! is about 1.85 × 10⁵⁰ times the age of the universe in seconds. That is why every shuffle of a 52-card deck is almost certainly a permutation no one has ever seen before in history — there are 52! possible orders, and humanity has shuffled only a tiny fraction of them.

Factorial Calculator Worked Examples

Every row below is computed exactly — hand-verified in Python — so you can check each one by hand.

The 52-Card Deck: 52P5 and 52C5

Deal 5 cards from a 52-card deck. If the order matters (a 5-card sequence), there are 52P5 = 52 × 51 × 50 × 49 × 48 = 311,875,200 ways. If the order does not matter (a poker hand), divide by the 5! = 120 orders of the same 5 cards: 52C5 = 311,875,200 / 120 = 2,598,960 possible 5-card hands. That single division by r! is the whole difference between permutations and combinations — and it is why 10P3 = 720 is exactly 6 times 10C3 = 120 (because 3! = 6).

The n!/(n−k)! Shortcut

You never need to compute two giant factorials to evaluate n!/(n−k)!. The bottom factorial cancels almost everything: 6!/(6−3)! = (6 × 5 × 4 × 3 × 2 × 1)/(3 × 2 × 1) = 6 × 5 × 4 = 120. Only the first k factors survive. The Expression mode shows this cancellation step by step — and it is also the permutation formula, since nPk = n!/(n−k)!.

How to Calculate a Factorial Step by Step

Follow this recipe for the classic example — calculating 5!.

1. Write the countdown: start at 5 and count down to 1: 5, 4, 3, 2, 1.
2. Multiply the first two: 5 × 4 = 20.
3. Multiply by the next: 20 × 3 = 60.
4. Keep going: 60 × 2 = 120.
5. Finish: 120 × 1 = 120. So 5! = 120.

Common Mistakes With Factorials

1. Saying 0! = 0

The most common factorial mistake: 0! is 1, not 0. It is the empty product, and the recurrence 1! = 1 × 0! forces it. If 0! were 0, every combination formula with r = 0 or r = n would divide by zero.

2. Claiming 52! beats the atom count

52! ≈ 8.066 × 10⁶⁷ is enormous — but the observable universe holds roughly 10⁸⁰ atoms, so 52! is about 8 × 10⁻¹³ of that: smaller, not bigger. Use the honest comparison instead: 52! is about 1.85 × 10⁵⁰ times the seconds since the Big Bang.

3. Mixing up nPr and nCr

Permutations (nPr) count orders; combinations (nCr) count sets. Arranging 3 books on a shelf is 10P3 = 720; choosing 3 books to take home is 10C3 = 120. Ask one question before you compute: "does the order matter?" If yes, nPr; if no, nCr.

4. Writing n! as n × (n−1) and stopping

n! is the product all the way down to 1, not just the first two factors. 5! is 5 × 4 × 3 × 2 × 1 = 120 — not 5 × 4 = 20. The countdown must reach 1 (or 0! = 1).

5. Expanding both factorials in n!/(n−k)!

Computing 52! and 47! separately to find 52P5 wastes effort and overflows most calculators. Cancel first: 52!/(52−5)! = 52 × 51 × 50 × 49 × 48 = 311,875,200. Only k factors survive.

6. Forgetting to divide by r! for combinations

10C3 is not 10 × 9 × 8 = 720 — that is 10P3. Combinations divide out the r! orders of the chosen items: 720/6 = 120. Forgetting the division overcounts by exactly r!.

Benefits and Limitations

Benefits

• Checks homework instantly: 5!, 0!, 10P3, 10C3 and 52! with every step shown.
• The 0! = 1 proof (empty product + recurrence + nC0 = 1) ends the most common confusion.
• The 52! scale section uses the correct comparison — smaller than ~10⁸⁰ atoms, vastly larger than the seconds since the Big Bang.
• Permutation vs combination is taught by the 6× gap (720 vs 120), not just stated.
• Plain-ASCII math notation renders cleanly in every browser and font.
• Responsive on phones, tablets and desktops.

Limitations

• Whole numbers only — factorials of fractions or negatives are not defined here (the gamma function extends them, but this tool does not).
• n is capped at 170 because 171! exceeds what JavaScript numbers can represent for the scientific-notation display; exact BigInt values are shown up to that cap.
• Very large exact values (like the 68 digits of 52!) are shown in full plus scientific notation for readability.

Quick Definitions

Related Math Tools

Factorial Calculator Methodology

This calculator derives every answer from the three standard counting formulas, using exact BigInt integer arithmetic (no floating-point rounding on the exact values shown).

Exact values are computed with BigInt multiplication from 1 to n, so 52! is produced digit-for-digit (80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000). Scientific notation is derived from the same exact value. The 52! scale comparison uses the commonly cited estimates of ~10⁸⁰ atoms in the observable universe and ~4.35 × 10¹⁷ seconds since the Big Bang (13.8 billion years); the ratio 52!/10⁸⁰ ≈ 8 × 10⁻¹³ and 52!/(4.35 × 10¹⁷) ≈ 1.85 × 10⁵⁰ are computed from those estimates, which are approximate. Inputs are restricted to whole numbers with 0 ≤ n ≤ 170 and 0 ≤ r, k ≤ n; anything else is rejected before any calculation. The page is intended as an educational calculator and verification tool.

Mathematical References

• Khan Academy – Factorial and Counting Seat Arrangements
• Khan Academy – Introduction to Permutations
• Math is Fun – Factorial (Definition)
• Wikipedia – Factorial