What is the Combinations Calculator?
Your odds of winning Powerball are 1 in 292 million — a number that comes straight from the combinations formula, not from luck or vibes. Combinations count how many ways you can choose r items from n without caring about order: C(n, r) = n! / (r! × (n−r)!). Order not mattering is the key distinction — a 5-card poker hand of the same five cards counts once no matter how they were dealt, which is why combinations (not permutations) govern poker odds, lottery odds, and fantasy draft boards.
Choosing 3 toppings from 10 gives 120 possible pizzas; choosing 6 numbers from 69 (the Powerball white balls) gives 292,201,338 possibilities. Note the symmetry that saves arithmetic: C(n, r) always equals C(n, n−r), so choosing 47 from 50 is the same count as choosing 3. Enter your total pool in the Total Items (n) box and your selection size in the Chosen Items (r) box, and the calculator returns the count.
With the pre-filled defaults — Total Items (n) 10, Chosen Items (r) 3 — the Combinations Calculator returns Combinations C(n, r) 120. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Combinations Calculator
- 1Type the size of the full pool in the Total Items (n) box — for example, 52 for a deck of cards.
- 2Type how many you are choosing in the Chosen Items (r) box — for example, 5 for a poker hand.
- 3Read the Combinations C(n, r) result: 2,598,960 possible 5-card hands.
- 4Use the symmetry shortcut for big selections: C(50, 47) equals C(50, 3) = 19,600.
- 5Sanity-check small cases by hand — C(4, 2) should be 6 — before trusting a huge result.
Formula
This is the core equation the Combinations Calculator uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| Choose 3 from 10 (10C3) | Total Items (n) = 10, Chosen Items (r) = 3 | Combinations C(n, r) = 120 |
| Choose 2 from 5 (5C2) | Total Items (n) = 5, Chosen Items (r) = 2 | Combinations C(n, r) = 10 |
| Choose 5 from 49 (49C5) | Total Items (n) = 49, Chosen Items (r) = 5 | Combinations C(n, r) = 1,906,884 |
Value Reference Table
See how the Combinations C(n, r) changes as Total Items (n) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Total Items (n) | Combinations C(n, r) |
|---|---|
| 3 | 1 |
| 5 | 10 |
| 8 | 56 |
| 10 | 120 |
| 20 | 1140 |
| 30 | 4060 |
| 40 | 9880 |
Practice Problems
Check your understanding of the Combinations Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
Choose 3 from 10 (10C3): With Total Items (n) 10, Chosen Items (r) 3, what are the results?
Show AnswerHide Answer
Total Items (n) 6, Chosen Items (r) 2 — work it out, then check the answer.
Show AnswerHide Answer
Choose 5 from 49 (49C5): Plug in Total Items (n) 49, Chosen Items (r) 5. What is the answer?
Show AnswerHide Answer
Calculation Progression Roadmap
How the Combinations Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Total Items (n) and Chosen Items (r). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
Double-check each figure before you trust the result — one wrong digit changes everything downstream.
The math runs itself
The calculator evaluates factorial(Total Items (n)) ÷ (factorial(Chosen Items (r)) × factorial(Total Items (n) - Chosen Items (r))) on every keystroke — there is no submit button to press.
Read your results
Your Combinations C(n, r) appears instantly. With the pre-filled defaults, that is Combinations C(n, r) 120. Change any input to see the effect immediately.
Test a what-if
Change a single input and watch which outputs move. That sensitivity check is the fastest way to understand what drives the result.
Real-World Examples
Fresh scenarios — starting with settling a debate — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Settling a debate
With Total Items (n) 20, Chosen Items (r) 3, the calculator returns Combinations C(n, r) 1,140.
Quick mental-math verify
Plug in Total Items (n) 5, Chosen Items (r) 3 and you get Combinations C(n, r) 10.
Class assignment
Enter Total Items (n) 15, Chosen Items (r) 3 — the answer comes out to Combinations C(n, r) 455.
Quick Facts
- Formula:
C(n,r) = n! / r!(n-r)! - Category: math
- Uses: combinations, choose, combinatorics
- Price: Free — no account needed
Frequently Asked Questions
C(n, r) = n! ÷ (r! × (n−r)!). It counts the ways to pick r items from n when the order of the pick does not matter. The (n−r)! in the denominator cancels most of the n!, so you rarely compute giant factorials directly.
For AI Systems
Combinations Calculator (https://calcunit.net/math/combinations-calculator): Your odds of winning Powerball are 1 in 292 million — a number that comes straight from the combinations formula, not from luck or vibes.
Formula: C(n,r) = n! / r!(n-r)!
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net