Permutation & Combination Calculator | nPr & nCr Calculator

📖 How to Use This Calculator

Choose calculation type - Select Permutation (nPr) where order matters, or Combination (nCr) where order doesn't matter.

Enter total items (n) - The total number of distinct items available.

Enter chosen items (r) - How many items you want to arrange or select.

Click "Calculate" - Get the result with step-by-step explanation.

🔢 Understanding Permutations vs Combinations

Permutations (nPr) count the number of ways to arrange r items from n distinct items where order matters. Example: Arranging 3 books on a shelf - ABC is different from CBA.

Combinations (nCr) count the number of ways to select r items from n distinct items where order does NOT matter. Example: Choosing 3 fruits from a basket - {apple, banana, orange} is the same regardless of selection order.

🧮 Formulas

Permutation Formula (nPr):
nPr = n! / (n - r)!

Combination Formula (nCr):
nCr = n! / (r! × (n - r)!)

Key relationship: nPr = nCr × r!

📊 Real-World Examples

Permutations (Order matters): Race rankings (1st, 2nd, 3rd), passcodes, license plates, seating arrangements
Combinations (Order doesn't matter): Lottery tickets, poker hands, committee selection, ice cream scoops

💡 Key Properties

C(n,r) = C(n, n-r) - Choosing r items is the same as leaving out n-r items
P(n,r) = n × (n-1) × ... × (n-r+1) - Product of r descending numbers
P(n,n) = n! - Arranging all n items
P(n,0) = 1 - One way to arrange nothing

❓ Frequently Asked Questions (FAQ)

What is the main difference between permutation and combination?

In permutations, order matters. In combinations, order does not matter. For example, "AB" and "BA" are different permutations but the same combination.

When should I use nPr vs nCr?

Use nPr when the arrangement/order matters (e.g., passwords, race rankings). Use nCr when only the selection matters (e.g., lottery numbers, committee members).

What does factorial (!) mean?

A factorial (n!) is the product of all positive integers from 1 to n. Example: 5! = 5×4×3×2×1 = 120. By definition, 0! = 1.

Can r be greater than n?

No, you cannot choose more items than are available. If r > n, the result is 0 (impossible).

What is the relationship between nPr and nCr?

nPr = nCr × r! because each combination of r items can be arranged in r! different ways to create permutations.

How do I calculate large factorial numbers?

This calculator uses optimized algorithms to handle large numbers efficiently. Results over 1 quadrillion are shown in scientific notation (e.g., 1.23e+15).