Calculate factorials, permutations, and combinations with exact integer arithmetic. See every digit when useful, or use digit counts, logarithms, scientific notation, and trailing zeros to understand results too large to scan.
Step by step
01Choose the calculation
Pick factorial for arrangements of all n items, permutations when order matters, or combinations when it does not.
02Enter n and r
Use non-negative integers, with r no larger than n for selections.
03Read the scale
The exact result appears with its digit count, scientific form, logarithms, parity, and trailing-zero count.
04Reveal or copy the integer
Long answers stay compact until you ask to show every digit.
Worked example
Given
Combination
10C3 = 10! / (3! × 7!)
= (10 × 9 × 8) / (3 × 2 × 1)
= 120The same ten people produce 720 ordered three-person lineups but only 120 committees, because each committee's six internal orders are the same selection.
Why this one
Order is the deciding question
ABC and BAC are distinct permutations but the same combination. Decide that before choosing nPr or nCr.
Zero factorial is one
0! = 1 keeps counting identities consistent: there is exactly one way to arrange nothing.
Exact integers matter quickly
Even 171! exceeds ordinary floating-point range. BigInt keeps the integer exact rather than returning infinity or rounded digits.
Trailing zeros come from pairs of 2 and 5
Factorials contain many more factors of 2, so counting factors of 5 determines how many decimal zeros n! ends with.
The judgement call
Arrange every item in a sequence
n!
Every item is used and order distinguishes outcomes.
Award gold, silver, and bronze
nPr
The same people in different places are different results.
Choose three committee members
nCr
Only membership matters, not internal order.
Choose with repetition allowed
Different formula
These nPr and nCr modes select without replacement.
Reference
FAQ
The factorial of a non-negative integer n is the product of every positive integer up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
There is one empty arrangement, and defining 0! as 1 makes the factorial recurrence and combination formulas work at their boundaries.
nPr counts selections where order matters; nCr counts groups where order does not. nPr is nCr multiplied by r!.
Yes. The calculator uses arbitrary-precision integers, so displayed digits are exact rather than floating-point approximations.
The trailing-zero count is the number of zeros at the right end of the exact integer, determined by the number of factors of 10 in the result.
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