combination calculator without repetition

What is a combination without repetition?

A combination without repetition answers this question: How many ways can I choose r items from n distinct items when order does not matter and each item can be picked at most once? This is also called the binomial coefficient, written as nCr or C(n, r).

For example, if you have 5 books and want to choose 2 to bring on a trip, selecting Book A then Book B is the same as selecting Book B then Book A. That is why this is a combination problem, not a permutation problem.

Combination formula (n choose r)

The standard formula is:

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

where ! means factorial. Factorial is the product of all positive integers up to that number (for example, 5! = 5 × 4 × 3 × 2 × 1).

Rules to remember

• n and r must be whole numbers.
• r cannot be greater than n.
• C(n, 0) = 1 and C(n, n) = 1.
• C(n, r) = C(n, n-r) (symmetry rule).

Quick examples

Example 1: Choosing 3 students from 10

C(10, 3) = 120. There are 120 unique groups of 3 students.

Example 2: Lottery style selection

In a 6-from-49 lottery, the number of possible tickets is C(49, 6) = 13,983,816. That large number explains why winning the jackpot is difficult.

Where combinations without repetition are used

• Lottery probability and odds
• Committee and team selection
• Card game analysis (like poker hands)
• Sampling in statistics
• Feature selection in data science

Combination vs permutation vs repetition

Combination (this page)

Order does not matter, and items are not reused.

Permutation

Order matters, and items are usually not reused.

Combination with repetition

Order does not matter, but items can be selected more than once.

Tips for accurate input

• Use non-negative integers only.
• If your question includes ranking or sequence, you probably need permutations.
• If one item can be selected multiple times, use combinations with repetition instead.

Final takeaway

The combination calculator without repetition is one of the most practical math tools for real-world counting problems. If your task is “choose r from n” and order does not matter, nCr is exactly what you need.