What is a factorial and how is it calculated?

A factorial of n (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely quickly: 10! = 3,628,800 and 20! exceeds 2 × 10^18.

What is 0! and why does it equal 1?

0! = 1 by mathematical convention. The reasoning: there is exactly one way to arrange zero objects (do nothing), which gives 0! = 1. This also keeps the recursive formula n! = n × (n-1)! consistent at n=1: 1! = 1 × 0! = 1 × 1 = 1. Without 0! = 1, many combinatorial formulas would break down.

What is the difference between permutation (nPr) and combination (nCr)?

Permutation nPr counts ordered arrangements: choosing 3 from 5 people for first, second, and third place gives 5P3 = 60. Combination nCr counts unordered selections: choosing 3 from 5 people for a committee gives 5C3 = 10. The relationship is nCr = nPr / r! because each unordered group of r items can be arranged r! ways.

What is the largest factorial this calculator can compute?

170! is the largest factorial representable in JavaScript double-precision floating point. 170! is approximately 7.257 × 10^306. The value 171! exceeds Number.MAX_VALUE (about 1.798 × 10^308) and evaluates to Infinity. For exact large factorials, specialized big-integer libraries are required.

How many ways can you arrange n objects?

The number of distinct orderings (permutations) of n objects is n!. For 3 objects (A, B, C): 3! = 6 arrangements (ABC, ACB, BAC, BCA, CAB, CBA). For 5 books on a shelf: 5! = 120 orderings. This is the full permutation nPn = n.

How is nCr used in the binomial theorem?

In the expansion of (a + b)^n, the coefficient of the term a^(n-k) × b^k is nCk (read: n choose k). For (a+b)^4: coefficients are C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1, giving 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4. The sum of all coefficients equals 2^n.

What does nCr equal when r = 0 or r = n?

nC0 = 1 for any n: there is exactly one way to choose zero items (choose nothing). nCn = 1 for any n: there is exactly one way to choose all n items (choose everything). These boundary values follow directly from the formula: n! / (0! × n!) = 1 and n! / (n! × 0!) = 1.

Factorial Calculator

Q: What is the formula for permutation nPr?

nPr = n! / (n-r)!. For 5P3: 5! / (5-3)! = 120 / 2 = 60. This counts ordered arrangements of r items chosen from n distinct items. The denominator cancels the factorial of items not chosen, leaving only the product of the top r terms: n × (n-1) × ... × (n-r+1).

Q: What is the formula for combination nCr?

nCr = n! / (r! × (n-r)!). For 5C3: 5! / (3! × 2!) = 120 / (6 × 2) = 10. The extra r! in the denominator (compared to permutation) divides out all orderings of the r selected items, leaving only the count of distinct groups.

Calculate n! factorials, nPr permutations, and nCr combinations for any positive integer up to 170.

🔢 What is a Factorial?

Factorial is the product of all positive integers from 1 up to a given number n. It is written as n! and pronounced "n factorial." For example, 5! = 5 × 4 × 3 × 2 × 1 = 120, and 10! = 3,628,800. By mathematical definition, 0! = 1 because there is exactly one way to arrange zero objects, and this definition keeps all combinatorial formulas consistent. The factorial function is defined only for non-negative integers.

Factorials appear throughout mathematics, statistics, and science. In combinatorics, n! gives the number of distinct orderings (permutations) of n objects: 3 books can be arranged on a shelf in 3! = 6 ways. In probability, factorials appear in binomial coefficients, which count the number of ways to choose k successes in n trials. In calculus, factorials appear in Taylor series and power series expansions: e^x = 1 + x + x²/2! + x³/3! + ...

This calculator covers three closely related computations. Factorial (n!) finds the product of all integers from 1 to n. Permutation (nPr) counts ordered arrangements: nPr = n! / (n-r)!, used when the order of selection matters (first place, second place, third place are distinct outcomes). Combination (nCr) counts unordered selections: nCr = n! / (r! × (n-r)!), used when only the membership of the group matters and not the order in which members were chosen.

A common source of confusion is whether to use permutation or combination. The key question is: does swapping two selected items create a different outcome? If yes (a lock combination where 1-2-3 differs from 3-2-1), use permutation. If no (a committee where the same three people form the same committee regardless of selection order), use combination. The relationship between them is nCr = nPr / r!, because dividing by r! removes the r! different orderings of the same group.

📐 Formula

n! = n × (n−1) × (n−2) × … × 2 × 1

n = non-negative integer (0 to 170 for this calculator)

0! = 1 (by definition)

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

nPr = n! ÷ (n − r)!

n = total number of distinct items

r = number of items to select (r ≤ n)

Example: 5P3 = 5! ÷ (5−3)! = 120 ÷ 2 = 60

nCr = n! ÷ (r! × (n − r)!)

n = total number of distinct items

r = number of items to select (r ≤ n)

Relationship: nCr = nPr ÷ r!

Example: 5C3 = 5! ÷ (3! × 2!) = 120 ÷ (6 × 2) = 10

📖 How to Use This Calculator

Steps

Select a mode using the tabs at the top: n! Factorial for a simple factorial, nPr Permutation for ordered arrangements, or nCr Combination for unordered selections.

Enter n in the first field (any whole number from 0 to 170). For permutation and combination modes, also enter r in the second field (must satisfy 0 ≤ r ≤ n).

Click Calculate to see the result. For n ≤ 15 in factorial mode, the full expansion (1 × 2 × … × n) is also shown.

💡 Example Calculations

Example 1 — Factorial of 7

Find 7! (how many ways can 7 runners finish a race?)

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

= 42 × 5 × 4 × 3 × 2 × 1 = 210 × 24 = 5,040

7! = 5,040

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Example 2 — Permutation 10P3 (top-3 finish from 10 runners)

How many ordered ways can 3 runners finish in first, second, and third place from a field of 10?

nPr = n! ÷ (n − r)! = 10! ÷ (10 − 3)! = 10! ÷ 7!

= (10 × 9 × 8 × 7!) ÷ 7! = 10 × 9 × 8 = 720

10P3 = 720 ordered podium finishes

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Example 3 — Combination 52C5 (5-card poker hands)

How many distinct 5-card hands can be dealt from a standard 52-card deck?

nCr = n! ÷ (r! × (n−r)!) = 52! ÷ (5! × 47!)

= (52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1) = 311,875,200 ÷ 120

52C5 = 2,598,960 distinct poker hands

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Example 4 — Combination 8C2 (pairs from a group)

How many pairs can be formed from a group of 8 people?

8C2 = 8! ÷ (2! × 6!) = (8 × 7) ÷ (2 × 1) = 56 ÷ 2

8C2 = 28 unique pairs

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❓ Frequently Asked Questions

What is a factorial in math?+

A factorial (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials count the number of ways to order n distinct objects. By convention, 0! = 1 because there is exactly one way to arrange zero items.

Why does 0 factorial equal 1?+

0! = 1 by definition. The most intuitive explanation: there is exactly one way to arrange zero objects, which is the empty arrangement. Mathematically, the definition keeps the recursive formula n! = n × (n-1)! valid at n=1: 1! = 1 × 0! = 1 × 1 = 1. Without this, formulas like nC0 = n! / (0! × n!) = 1 would fail.

What is the difference between permutation and combination?+

Permutation (nPr) counts ordered selections: choosing 3 gold-silver-bronze medalists from 10 athletes gives 10P3 = 720 because the order (who gets gold vs. silver) matters. Combination (nCr) counts unordered selections: choosing 3 committee members from 10 gives 10C3 = 120 because the group is the same regardless of selection order. nCr = nPr / r! because each group of r items can be ordered r! ways.

What is 10 factorial?+

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800. This is the number of ways to arrange 10 distinct items in a line. For context: a standard deck of 52 cards can be shuffled in 52! arrangements, a number with 68 digits that vastly exceeds the number of atoms in the observable universe.

What is the formula for permutation nPr?+

nPr = n! / (n-r)!. For 8P3: 8! / 5! = (8 × 7 × 6 × 5!) / 5! = 8 × 7 × 6 = 336. The denominator cancels the shared factorial tail, leaving only the top r terms of n!. This is also written P(n,r) or P_n^r in some textbooks.

What is the formula for combination nCr?+

nCr = n! / (r! × (n-r)!). For 8C3: 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56. The extra r! divides out the orderings of the selected items, leaving the count of distinct subsets. Also written C(n,r) or the binomial coefficient "n choose r."

What is the largest factorial I can calculate here?+

This calculator supports up to 170!. The value 170! is approximately 7.257 × 10^306, which is the largest factorial that fits in a JavaScript double-precision floating-point number. Values from about 21! onwards are returned in scientific notation (e.g., 4.2749e+19) because they exceed JavaScript's integer precision limit of 2^53.

How many ways can you arrange 5 objects?+

5! = 120 ways. The first position can be filled by any of the 5 objects, the second by any of the remaining 4, and so on: 5 × 4 × 3 × 2 × 1 = 120. More generally, n distinct objects can be arranged in n! ways, which is the full permutation nPn = n!.

Is nCr symmetric? Does 10C3 equal 10C7?+

Yes. nCr = nC(n-r) for any valid r. For 10C3 = 10C7: both equal 120. The reason: choosing 3 items to include from 10 is equivalent to choosing 7 items to exclude. This symmetry appears visually in Pascal's triangle, where every row reads identically from both ends.

How is factorial used in probability and statistics?+

Factorials are central to combinatorics and probability. The binomial probability formula P(X=k) = nCk × p^k × (1-p)^(n-k) uses nCk to count favorable outcomes. Normal distribution approximations use Stirling's approximation for large n! (ln(n!) = n ln n - n + 0.5 ln(2πn)). Poisson distribution also involves factorials in its pmf: P(k) = e^(-λ) × λ^k / k!.

What is 52 choose 5 (number of poker hands)?+

52C5 = 52! / (5! × 47!) = 2,598,960. This is the total number of distinct 5-card poker hands from a 52-card deck. Of these, 4 are royal flushes, 36 are straight flushes, 624 are four-of-a-kind hands, and so on. Poker hand probabilities are calculated by dividing each hand count by 2,598,960.

Tags: Factorial Calculator Factorial N Factorial Permutation Calculator Combination Calculator NPr NCr Combinatorics Calculator Formula Free Online

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📌 Quick Tips

💡0! = 1 by definition. This makes combinatorial formulas consistent: nC0 = 1 (one way to choose nothing) and nP0 = 1 work correctly only when 0! = 1.

💡For permutations nPr, order matters: selecting A then B is a different arrangement from B then A. Use combinations nCr when the order of selection does not matter.

💡The factorial grows extremely fast: 10! = 3,628,800 and 20! = 2,432,902,008,176,640,000. Values above 170! exceed JavaScript floating-point range and return scientific notation.

💡nCr = nC(n-r): choosing r items to include gives the same count as choosing n-r items to exclude. So 10C3 = 10C7 = 120.

💡The sum of all nCr values for a fixed n equals 2^n. For n = 4: C(4,0)+C(4,1)+C(4,2)+C(4,3)+C(4,4) = 1+4+6+4+1 = 16 = 2^4.