If you have ever tried to calculate the odds of winning the lottery, or figure out how many different 5-card poker hands you can draw from a deck, you are doing combinatorics math. In these scenarios, two specific rules apply: the order you draw the items in does not matter, and once an item is drawn, it cannot be drawn again. In mathematics, this is known as a Combination Without Repetition.
Our free online Combination Without Repetition Calculator instantly solves this massive algebraic equation. Also known as an “nCr Calculator”, this tool quickly evaluates your total pool of items and how many you are choosing, outputting the exact number of possible groups that can be formed.
Understanding the nCr Formula
To calculate a combination without repetition, the calculator uses the standard nCr Formula: C(n,r) = n! / [r! × (n - r)!]. Because this formula uses factorials (the exclamation point), doing the math by hand can take hours. Here is exactly what those variables mean.
| Equation Variable | Statistical Definition | Real-World Example (Playing Poker) |
|---|---|---|
| The n Variable | The Total Pool Size | The total number of items you have available to pick from. In poker, the n variable is exactly 52 (the number of cards in a deck). |
| The r Variable | The Number Chosen | How many items you are physically picking out of the pool. In 5-Card Draw poker, the r variable is exactly 5. |
| The ! Symbol | The Factorial | This tells the calculator to multiply the number by every integer below it. (e.g., 5! = 5 × 4 × 3 × 2 × 1). This mathematical trick is what calculates the massive “without repetition” odds. |
Combinations vs. Permutations (The Order Rule)
The most common mistake students make is using a Combination formula when they actually need a Permutation formula. The entire difference boils down to a single question: Does the order matter?
| Mathematical Concept | Does Order Matter? | Real-World Example |
|---|---|---|
| Combination | NO | Making a fruit salad. If you pick Apples, Bananas, and Grapes, it does not matter what order you chop them in. It is the exact same salad. |
| Permutation | YES | Entering a PIN code. The numbers 1-2-3-4 will unlock your phone, but the numbers 4-3-2-1 will not. The specific order is strictly enforced. |
If the order of your items is strictly enforced (like a race where 1st, 2nd, and 3rd place matter), you must switch to our Permutation Without Repetition Calculator. If you are calculating the probability of a sequence of events happening, use our AND Probability Calculator.
Frequently Asked Questions (FAQ)
What does “Without Repetition” mean?
It means that once you pull an item out of the pool, you cannot put it back. If you are drawing cards from a deck, once you draw the Ace of Spades, it is physically impossible to draw the Ace of Spades a second time. The total pool size (n) shrinks by 1 after every single pick.
Is a combination lock actually a combination?
No! This is a massive joke among mathematicians. Because the order of a combination lock strictly matters (e.g., 30-15-05 will open the lock, but 05-30-15 will not), it is mathematically a Permutation Lock. The English language completely botched the terminology.
Why are Combination numbers always smaller than Permutation numbers?
Because combinations do not care about order. In a permutation, the group “A-B-C” and the group “C-B-A” are counted as two completely different, unique results. In a combination, they contain the exact same three letters, so they are grouped together and only counted as one single result. This drastically shrinks the final number.