组合计算器 (nCr)

This Combinations Calculator (nCr) helps you compute the total number of ways to choose a sample size from a set of distinct objects where order does not matter and repetitions are not allowed. It’s perfect for solving problems involving combinations and permutations in probability, statistics, and more.

What Are Combinations?

In combinatorics, a combination is a way of selecting items from a larger set where the order is not important. This differs from permutations, where the order is important.

The standard combinations formula is:

nCr = n! / [r!(n - r)!]

Where:

n = total number of elements in the set

r = sample size or number of items selected

! = factorial

This calculator computes combinations without repetitions — meaning each object is chosen only once per combination.

When to Use This Calculator

Selecting a group of winners from a larger pool

Choosing cards from a deck where order does not matter

Solving statistical problems related to combinations and permutations

Calculating the total number of permutations when only combinations are required

How It Works

Enter the number of elements (n): Input the total count in the set of objects.

Enter the sample size (r): Define how many elements you want to choose.

Click Calculate: The calculator uses the combinations formula to compute the result.

View the result: You’ll see how many ways you can choose r elements from n when order does not matter.

Example Calculation

Suppose you want to choose 3 elements from a set of 10:

n = 10, r = 3

10C3 = 10! / (3!(10-3)!) = (10 × 9 × 8) / (3 × 2 × 1) = 120

So, there are 120 combinations of 3 elements from a set of 10.

Frequently Asked Questions

1. What’s the difference between combinations and permutations?

Combinations are used when order does not matter, while permutations apply when the order is important. For example, selecting team members is a combination, while assigning tasks is a permutation.

2. Can I calculate combinations with repetitions?

This calculator is designed for combinations without repetitions. If repetitions are allowed, a different formula must be used: n+r-1Cr.

3. What happens if the sample size exceeds the total number of elements?

You cannot select more elements than are available in the set. If r > n, the combination is mathematically undefined.