Permutation & Combination Calculator

nPr and nCr for a given set size and selection size.

Overview

Calculates Permutations nPr (where order matters) and Combinations nCr (where order does not matter) for choosing r items from n distinct items.

Formula & Calculation Method

Permutations: nPr = n! / (n - r)! | Combinations: nCr = n! / [r! × (n - r)!]

Permutations count ordered arrangements (passwords, race podiums); combinations count unordered groups (lottery picks, card hands).

Worked Example: Selecting 3 Committee Members from 8 Candidates (n=8, r=3)

Inputs: Total items n = 8 | Chosen items r = 3

  • Combination formula: 8! / [3! × (8 - 3)!] = 8! / [3! × 5!]
  • Expand: (8 × 7 × 6 × 5!) / [(3 × 2 × 1) × 5!]
  • Cancel 5!: (8 × 7 × 6) / 6 = 336 / 6 = 56 combinations
  • If order mattered (Permutation nPr): 8 × 7 × 6 = 336 permutations

Summary: n = 8, r = 3 | Combinations nCr = 56 | Permutations nPr = 336

Common Use Cases

  • Lottery & Gaming Odds: Calculate the total number of possible combinations in 6/49 lottery games.
  • Team Selection: Determine how many unique project groups can be selected from a department pool.
  • Password Security: Calculate the total combinatorial search space required to brute-force security pins.

Frequently Asked Questions

What is the key difference between permutation and combination?

Order. In a permutation, arrangement matters (ABC is different from CBA); in a combination, grouping matters (ABC and CBA are the exact same group).

Why is a 'combination lock' a misnomer?

Because the order of digits matters (e.g. 1-2-3 is not 3-2-1), making it mathematically a permutation lock.

What is Pascal's Triangle in combinations?

A triangular array where each entry corresponds to a combination nCr, with rows representing n and columns representing r.

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