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Permutation & Combination Calculator

Calculate nPr and nCr values instantly

Result: nPr
0
n! / (n – r)!

Understanding Permutations and Combinations

P
Permutation (nPr)
Number of ways to arrange ‘r’ items from ‘n’ items where ORDER matters. Example: Arranging 2 players from 11 as captain and goalkeeper.
C
Combination (nCr)
Number of ways to choose ‘r’ items from ‘n’ items where ORDER doesn’t matter. Example: Choosing 2 strikers from 11 team members.
Formulas:
Permutation
nPr = n! / (n – r)!
Combination
nCr = n! / (r! × (n – r)!)

Understanding Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics that deals with counting and arranging objects. While both deal with selections from a set, they differ in whether the order of selection matters.

Permutations

Permutations refer to the arrangements of elements where the order is important. For example, if you have a combination lock, the sequence in which you enter the numbers matters.

Key Points:

  • Order Matters: In permutations, changing the order of selected items creates a different arrangement.
  • Without Replacement: This means once an item is chosen, it cannot be chosen again.

Formula for Permutations: The number of ways to arrange ( r ) elements from a set of ( n ) elements is given by:

[ nPr = \frac{n!}{(n – r)!} ]

Where:

  • ( n! ) (n factorial) is the product of all positive integers up to ( n ).
  • ( r ) is the number of elements being arranged.

Example: Imagine you have a soccer team of 11 players (A, B, C, D, E, F, G, H, I, J, K) and you want to choose a captain and a goalkeeper.

  1. Choose the Captain: You have 11 options (A to K).
  2. Choose the Goalkeeper: After choosing the captain, you have 10 remaining options.

So, the total number of ways to choose a captain and a goalkeeper is:

[ 11P2 = 11 \times 10 = 110 ]

Combinations

Combinations refer to selections of elements where the order does not matter. For instance, if you are choosing two strikers from the same soccer team, it doesn’t matter if you pick A first and then B, or B first and then A; they are still the same pair.

Key Points:

  • Order Does Not Matter: In combinations, the arrangement of selected items is irrelevant.
  • Without Replacement: Similar to permutations, once an item is chosen, it cannot be chosen again.

Formula for Combinations: The number of ways to choose ( r ) elements from a set of ( n ) elements is given by:

[ nCr = \frac{n!}{r! \times (n – r)!} ]

Where:

  • ( r! ) accounts for the redundancies in arrangements since order does not matter.

Example: Using the same soccer team, if you want to choose 2 strikers from 11 players:

  1. The total arrangements (permutations) of choosing 2 players is ( 11P2 = 110 ).
  2. Since the order does not matter, you divide by the number of ways to arrange 2 players (which is ( 2! = 2 )):

[ 11C2 = \frac{11!}{2! \times (11 – 2)!} = \frac{11!}{2! \times 9!} = \frac{110}{2} = 55 ]

Summary

  • Permutations: Order matters. Use the formula ( nPr = \frac{n!}{(n – r)!} ).
  • Combinations: Order does not matter. Use the formula ( nCr = \frac{n!}{r! \times (n – r)!} ).