Possible Combinations Calculator

Calculate combinations with and without repetition using statistical formulas

Calculate Possible Combinations

The total number of distinct objects available

The number of objects to choose from the total

Each object can only be chosen once (no duplicates)

Example: Combinations of 5 Letters

Problem

Question: How many combinations of 5 letters are possible from the English alphabet?

Given: n = 26 (total letters), r = 5 (letters to choose)

Solution

Without repetition: C(26,5) = 26! / (5!(26-5)!) = 65,780

With repetition: C'(26,5) = (5+26-1)! / (5!(26-1)!) = 142,506

Formula Reference

Without Repetition

C(n,r) = n! / (r!(n-r)!)

Each object can only be chosen once. Order doesn't matter.

With Repetition

C'(n,r) = (r+n-1)! / (r!(n-1)!)

Objects can be chosen multiple times. Order doesn't matter.

Key Concepts

C

Combinations

Selection where order doesn't matter

P

Permutations

Arrangement where order matters

!

Factorial

Product of all positive integers up to n

Quick Tips

Use combinations when order doesn't matter

With repetition allows choosing the same element multiple times

C(n,0) = 1 for any n ≥ 0

C(n,r) = C(n,n-r)

Understanding Combinations

What are Combinations?

Combinations are ways to select items from a collection where the order of selection doesn't matter. This is different from permutations where order is important.

When to Use Combinations

  • Selecting team members from a group
  • Choosing lottery numbers
  • Selecting items from a menu
  • Creating subsets from a set

Combinations vs. Permutations

Example: Choosing 2 letters from (A, B, C)

Combinations (order doesn't matter)

  • (A, B)
  • (A, C)
  • (B, C)

Total: 3

Permutations (order matters)

  • AB, BA
  • AC, CA
  • BC, CB

Total: 6

Repetition Explained

Without repetition: Each element can only be chosen once (like drawing cards without replacement)
With repetition: Elements can be chosen multiple times (like drawing cards with replacement)