Permutation Calculator

How it works

Calculate Permutations (nPr) and Combinations (nCr).

Solve for permutations (nPr) and combinations (nCr)

Description

The Permutation (and Combination) Calculator is an essential tool for “enumerative combinatorics”—the math of counting possible arrangements. It helps you determine how many ways you can select a subset of items from a larger group. It provides two distinct results: Permutations (where the order of selection creates a different outcome) and Combinations (where only the selection group matters, regardless of order).

Inputs

• Total Items (n): The size of the pool you are selecting from.
• Items to Choose (r): The number of items you are actually picking.

Outputs

• Permutations (nPr): Total arrangements where order matters (e.g., a 1st, 2nd, and 3rd place race).
• Combinations (nCr): Total groups where order does not matter (e.g., picking a committee of 3 people).

Chart

• N/A: This tool provides categorized counting results.

“Good to Know”

• “Order Matters” Example: For the letters A and B, there are 2 permutations (AB, BA) but only 1 combination (the group containing both letters).
• Formula (nPr): $n! / (n-r)!$
• Formula (nCr): $n! / (r! \cdot (n-r)!)$
• Exponential Growth: The number of arrangements grows incredibly fast as $n$ increases. A standard 52-card deck has more possible permutations than there are atoms in the solar system!

Examples

Example 1: Picking a Committee

• Input: Total (n): 10, Choose (r): 3
• Output:
  • Permutations: 720
  • Combinations: 120 (Choosing a 3-person team from 10)

Example 2: Assigning Race Winners

• Input: Total (n): 8, Choose (r): 3
• Output:
  • Permutations: 336 (Ways for 8 runners to take Gold, Silver, and Bronze)

Example 3: Small Pair

• Input: Total (n): 3, Choose (r): 2
• Output:
  • Permutations: 6
  • Combinations: 3