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Permutations and Combinations calculator for nPr and nCr. Calculate permutations, combinations, formulas, and results quickly online.
| Equation | Meaning | Use When |
|---|---|---|
| n! = n(n−1)(n−2)···2×1 | Factorial | 0! = 1 |
| nPr = n! ÷ (n−r)! | Permutation | Order matters |
| nCr = n! ÷ (r!(n−r)!) | Combination | Order doesn't matter |
| nPr = nCr × r! | Relationship | Each group of r has r! orderings |
| nCr = nC(n−r) | Symmetry identity | Reduces calculation work |
Our Permutations and Combinations calculator helps you solve counting problems in seconds. You can use it to calculate permutations, combinations, or compare both results.
The calculator uses two main values: n and r. Here, n is the total number of objects, while r is the number of objects you select.
Use permutations when order matters. Use combinations when order does not matter. The calculator uses direct product loops, so it can avoid large factorial calculations when possible.
The permutation formula is:
nPr = n! / (n − r)!
It can also be written as:
nPr = n × (n − 1) × (n − 2) × ... × (n − r + 1)
A permutation counts different orders as different results.
For example, choosing A, B, and C gives different arrangements such as ABC, ACB, and BAC.
The combination formula is:
nCr = n! / [r! × (n − r)!]
A combination does not treat different orders as new results. So, choosing A, B, and C is the same group as C, A, and B.
The calculator uses a shorter calculation when possible:
rr = min(r, n − r)
This uses the smaller value between r and n − r. It makes combination calculations more efficient.
The key relationship is:
nPr = nCr × r!
This lets you convert a combination into a permutation when the selected objects can be arranged in different orders.
Both n and r must be non-negative whole numbers. Also, r cannot be greater than n.
Suppose you have 5 different books and want to arrange 3 of them.
Here:
n = 5
r = 3
Because the order of the books matters, use permutations.
nPr = n! / (n − r)!
5P3 = 5! / (5 − 3)!
5P3 = 5! / 2!
5P3 = 120 / 2
5P3 = 60
So, there are 60 possible arrangements.
Now suppose you want to choose 3 books from the same 5 books, but their order does not matter.
Use combinations:
nCr = n! / [r! × (n − r)!]
5C3 = 5! / (3! × 2!)
5C3 = 120 / (6 × 2)
5C3 = 10
So, there are 10 possible groups.
You can also verify the relationship:
nPr = nCr × r!
60 = 10 × 3!
60 = 10 × 6
60 = 60
This confirms the permutation and combination relationship.
Our Permutations and Combinations calculator makes common counting calculations fast and simple. It calculates nPr when order matters and nCr when order does not matter.
It can also compare both values and show the relationship between them. Whether you're checking a math problem, studying probability, or working with counting formulas, the calculator gives you a quick way to find the correct result.
It is an online math tool that calculates permutations and combinations from n and r values. It can show nPr, nCr, and their relationship.
A permutation is used when order matters. A combination is used when order does not matter.
The formula is:
nPr = n! / (n − r)!
The formula is:
nCr = n! / [r! × (n − r)!]
No. For this calculator, r must be less than or equal to n. You cannot select more objects than the total number available.
Yes. Our online Permutations and Combinations calculator lets you enter n and r and calculate the required result without doing the full calculation by hand.
The mathematical formulas are the same. A TI-84 can calculate nPr and nCr, while an online calculator can provide a simpler interface and calculation details.
Yes. Excel has functions for permutations and combinations. However, an online calculator can be more convenient when you want a quick result and formula-based explanation.
n represents the total number of available objects, items, people, or choices.
r represents the number of objects selected or arranged from the total n objects.