Factorial Calculator

Calculate n!, nPr (permutations), and nCr (combinations) instantly.

n! Factorial Calculator
n!
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📖 What is a Factorial?

A factorial is denoted by an exclamation mark (n!) and represents the product of all positive integers from 1 up to n. The factorial function appears throughout mathematics - in combinatorics, probability theory, number theory, and calculus.

For example: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. This tells us there are 720 ways to arrange 6 different objects in a sequence.

Factorials are fundamental to two key counting techniques - permutations and combinations - which are used any time you need to count the number of possible arrangements or selections from a group.

Permutations (nPr) answer the question: *In how many ways can I arrange r items from a group of n, where the order matters?* For example, the number of ways 3 runners can finish first, second, and third from a group of 10 athletes is 10P3 = 720.

Combinations (nCr) answer: *In how many ways can I choose r items from n, where order doesn't matter?* For example, the number of ways to choose 3 people for a committee from a group of 10 is 10C3 = 120 - much less than 720, because the same group of 3 people is only counted once regardless of the order they were chosen.

📐 Formula

n! = n × (n−1) × (n−2) × ... × 2 × 1
0! = 1 (by definition)
nPr = n! / (n−r)! [Permutation - order matters]
nCr = n! / (r! × (n−r)!) [Combination - order doesn't matter]

📖 How to Use This Calculator

1
Select the mode: n! Factorial, nPr Permutation, or nCr Combination.
2
Enter n - the total number of items.
3
For permutation and combination, also enter r - the number of items to select.
4
Click Calculate - the result and formula breakdown are shown.

💡 Example Calculations

Example 1 - Arranging books

1
How many ways can you arrange 8 books on a shelf?
2
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 ways
Try this example →

Example 2 - Lottery combination

1
In a lottery, you pick 6 numbers from 1 to 49. How many possible tickets are there?
2
49C6 = 49! / (6! × 43!) = 13,983,816 combinations
3
Each ticket has a 1-in-13.98-million chance of winning the jackpot.
Try this example →

Example 3 - Podium positions

1
In a race with 12 competitors, how many ways can the top 3 podium spots be filled?
2
12P3 = 12! / (12−3)! = 12 × 11 × 10 = 1,320 arrangements
Try this example →

Frequently Asked Questions

What is a factorial?+
A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials appear in permutations, combinations, probability, and many areas of mathematics.
What is 0 factorial?+
0! = 1. This is defined by mathematical convention and is essential for the combinatorial formulas to work correctly. It represents the one way to arrange zero objects - by doing nothing.
What is the difference between nPr and nCr?+
nPr (permutation) counts ordered arrangements: how many ways to choose r items from n where order matters. nCr (combination) counts unordered selections: how many ways to choose r items from n where order doesn't matter. nCr = nPr ÷ r!
What is the largest factorial this calculator can compute?+
This calculator handles factorials up to 170! accurately. Beyond that, JavaScript's floating-point numbers overflow to Infinity. For extremely large factorials, Stirling's approximation is used in advanced mathematics.
Where are factorials used in real life?+
Factorials appear in probability (how many outcomes are possible), statistics (permutation and combination tests), cryptography (key space calculations), and computer science (algorithm complexity analysis). Shuffling a deck of cards has 52! ≈ 8×10⁶⁷ possible arrangements.
What is the difference between a permutation and a combination?+
A permutation counts ordered arrangements - the order matters. A combination counts unordered selections - the order does not matter. Example: selecting 2 students from 4 (A, B, C, D). Permutations (ordered): AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC = 12 arrangements. Combinations (unordered): AB, AC, AD, BC, BD, CD = 6 selections. nPr = n! / (n-r)! and nCr = n! / (r! x (n-r)!).
What is 0! (zero factorial)?+
0! = 1 by definition. This is not just a convention - it is mathematically necessary for formulas involving permutations and combinations to work correctly. For example, nCr when r = 0 or r = n must equal 1 (there is exactly one way to choose none or all items). If 0! were 0, these formulas would break down. The result follows from the recursive definition: n! = n x (n-1)!, so 1! = 1 x 0!, giving 0! = 1.
How are factorials used in probability?+
Factorials are the foundation of counting in probability. They appear in: (1) Combinations: nCr = n! / (r! x (n-r)!) - used to find the probability of selecting k items from n. (2) Permutations: nPr = n! / (n-r)! - used when order matters. (3) The binomial distribution formula. (4) Calculating odds in card games, lottery probabilities, and combinatorial problems. Example: the probability of being dealt a specific 5-card poker hand uses combinations from 52 cards.
Tags: Factorial N Factorial Permutation Combination N! Combinatorics Probability NPr NCr

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📌 Quick Tips

💡0! = 1 by definition - this is a mathematical convention, not an error.
💡Factorials grow incredibly fast: 10! = 3.6 million, 20! = 2.4 quintillion. Numbers above 170! overflow standard floating point.
💡Use nCr when the order of selection doesn't matter (e.g. choosing a committee). Use nPr when order matters (e.g. race rankings).