Math

nCr Calculator

Calculate n Choose r Instantly

Enter any values for n and r to compute the binomial coefficient C(n,r) = n!/(r!(n-r)!). The calculator shows the full step-by-step substitution, an interactive Pascal's triangle where you can click any cell to fill in n and r, and a distribution chart of all C(n,r) values for fixed n. Supports exact BigInt arithmetic for n up to 10,000.

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Lottery Draw

Calculate the number of possible 6/49 lottery combinations.

Key values: Total balls: 49 · Drawn: 6 · Combination mode

Race Podium

Find the number of ways to award Gold, Silver, and Bronze among 8 runners.

Key values: Runners: 8 · Medals: 3 · Permutation mode

Secret Santa

Calculate valid gift assignments where nobody draws their own name.

Key values: Participants: 8 · Derangement mode

Password Keyspace

Estimate the number of possible 8-character passwords from printable ASCII.

Key values: Character set: 94 · Length: 8 · Repetition allowed

Documentation

This calculator is also known as nCr Calculator.

Read the complete guide

What Is nCr?

nCr (also written as C(n,r) or "n choose r") counts the number of ways to select r items from n distinct items when order does not matter. It is the binomial coefficient that appears in polynomial expansions and probability distributions.

How to Calculate nCr

Follow these steps to compute C(n,r):

Write the formula: C(n,r) = n! / (r! * (n-r)!)
Use the smaller of r or n-r to minimize computation
Cancel common factorial terms to avoid overflow
The result is always a whole number

Examples

Lottery: C(49, 6)

How many possible 6/49 lottery ticket combinations exist?

C(49, 6) = 49!/(6! * 43!) = 13,983,816 possible combinations.

Key takeaway: Your chance of winning is about 1 in 14 million.

Using nCr Effectively

Key strategies for computing binomial coefficients:

Use the symmetry C(n,r) = C(n,n-r) to pick the smaller side and speed up computation
Click cells in the interactive Pascal's triangle to explore coefficient patterns
For large n, the calculator uses BigInt arithmetic for exact results

Frequently Asked Questions about nCr Calculator

What is the difference between nCr and nPr?

nCr (combinations) ignores order -- {A,B} and {B,A} are the same. nPr (permutations) counts order -- AB and BA are different. The relationship is nPr = nCr * r!.