Math

n Choose r Calculator

How Many Ways Can You Choose?

Type in the total number of items (n) and how many you want to pick (r), and this calculator instantly tells you how many different selections are possible. Perfect for homework problems, probability questions, and real-world counting problems like lottery odds or committee selection.

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 n Choose r Calculator.

Read the complete guide

Understanding "n Choose r"

"n choose r" is everyday language for the binomial coefficient C(n,r). It answers the question: how many ways can I select r items from a group of n?

Examples

Committee of 3 from 12

How many 3-person committees can be formed from 12 candidates?

C(12, 3) = 220 possible committees.

Key takeaway: Since committee members have no ranked roles, order does not matter.

Solving "How Many Ways" Problems

Approach counting problems systematically:

Identify n (total items) and r (items selected)
Determine whether order matters (permutation vs. combination)
Check whether repetition is allowed

Frequently Asked Questions about n Choose r Calculator

How do I know whether order matters?

Ask yourself: would rearranging my selection give a different outcome? If picking {A,B} is the same as {B,A}, use combinations (n choose r). If the order matters (like 1st vs 2nd place), use permutations.

What are C(n, 0) and C(n, n)?

Both equal 1. C(n,0) = 1 because there is exactly one way to select nothing. C(n,n) = 1 because there is only one way to select every item. These are the first and last entries in every row of Pascal's triangle.

Is "n choose r" the same as the binomial coefficient?

Yes. "n choose r" is another name for the binomial coefficient, written C(n,r) or ⁿCᵣ. It appears in Pascal's triangle and gives the coefficient of each term in the binomial theorem expansion of (a+b)^n.