Permutations & Combinations Calculator - nPr and nCr
Our free permutations and combinations calculator helps you solve complex counting problems instantly. Whether you need to calculate nPr (permutations)or nCr (combinations), this tool provides precise results and shows the formulas used. Perfect for probability studies, statistics homework, and mathematical research.
Key Features of Our nPr and nCr Calculator
- 🔢 Dual Calculation: Get both permutation and combination results at once
- 📊 Formula Display: See exactly how nPr and nCr are calculated for your numbers
- ⚡ Real-time Results: Results update instantly as you change input values
- 📐 Mathematics Focus: Clear explanation of order and selection rules
- 📱 Simple Interface: Easy to use on desktops, tablets, and smartphones
Permutation (nPr) Formula
A permutation is used when the arrangement of items matters. The mathematical formula is:P(n, r) = n! / (n - r)!
Where n is the total number of items and r is the number of items being arranged.
Combination (nCr) Formula
A combination is used when only the selection matters, regardless of order. The mathematical formula is:C(n, r) = n! / [r! * (n - r)!]
Because order doesn't matter, we divide by r! to eliminate the different arrangements of the same items.
Common Use Cases
Permutations Examples
- • Phone numbers and postal codes
- • Lock combinations (which are actually permutations!)
- • Podium finishers in a marathon
- • Words formed using specific letters
Combinations Examples
- • Lottery number selections
- • Choosing members for a committee
- • Card games like Poker and Bridge
- • Picking fruit for a salad
Frequently Asked Questions
What is the difference between permutation and combination?
The main difference is order. In a permutation, the order of items matters (like a combination lock code). In a combination, the order does not matter (like a fruit salad or a hand of cards).
What does nPr mean?
nPr stands for the number of permutations of n items taken r at a time. The formula is n! / (n-r)!. It calculates the number of ways to arrange r objects from a set of n objects.
What does nCr mean?
nCr stands for the number of combinations of n items taken r at a time. The formula is n! / (r!(n-r)!). It calculates the number of ways to choose r objects from a set of n, where order is not important.
How do I calculate factorials for nPr and nCr?
A factorial (indicated by !) is the product of all positive integers up to that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in the denominators of these formulas to remove duplicate arrangements.
When should I use permutations?
Use permutations when you are dealing with arrangements, sequences, codes, seating charts, or finishing positions in a race—any scenario where first, second, and third are distinct.
When should I use combinations?
Use combinations when you are forming committees, picking a group of players for a team, selecting toppings for a pizza, or drawing cards—any scenario where the group composition is all that matters.