Imagine you are walking into a bakery to buy a dozen donuts. The bakery offers exactly 5 different flavors. You can choose any combination you want, and because they have an infinite supply, you can easily pick the exact same flavor multiple times. Furthermore, the order the donuts are placed in the box does not matter at all. In discrete mathematics, this exact scenario is known as a Combination with Repetition (or Combination with Replacement).
Our free online Combinations with Repetition Calculator instantly solves these massive “multichoose” probability questions. By evaluating your total categories (the donut flavors) and your total selections (the dozen donuts), the calculator outputs the exact number of possible unique combinations you can physically create.
Understanding the “With Repetition” Formula
To calculate a combination where repetition is allowed, the formula must mathematically expand to account for duplicate picks. The formula is: C(n, r) = (n + r – 1)! / [r! × (n – 1)!]. Here is how those variables break down.
| Equation Variable | Statistical Definition | Real-World Example (The Bakery) |
|---|---|---|
| The n Variable | The Total Categories | The total number of types of items you can pick from. If the bakery has 5 flavors of donuts, your n is exactly 5. |
| The r Variable | The Total Selections | How many total items you are physically placing in your box. If you are buying a dozen donuts, your r is exactly 12. |
| The – 1 Modifier | The Dividers | Mathematically, to separate 5 flavors of donuts, you only need 4 “dividers” between them. The n – 1 accounts for these category dividers. |
With Repetition vs. Without Repetition
The single most common mistake in combinatorics is using the wrong formula for your physical scenario. The difference relies entirely on whether your “pool” of items shrinks when you pull an item out.
| Mathematical Concept | Can I pick the same item twice? | Real-World Example |
|---|---|---|
| With Repetition (Replacement) | YES | Picking 3 scoops of ice cream from 10 available flavors. You can easily choose 3 scoops of pure Vanilla. The pool of vanilla does not “disappear.” |
| Without Repetition | NO | Drawing 5 cards from a standard deck. Once you draw the Ace of Spades, it is physically impossible to draw it again. The pool shrinks. |
If your scenario physically prevents you from picking the same item twice (like a lottery drawing or a card game), you must switch to our standard Combination Without Repetition Calculator. If the exact order of your items is strictly enforced (like a padlock code), you need to use our Permutation With Repetition Calculator.
Frequently Asked Questions (FAQ)
What does “With Replacement” mean?
In statistics, “With Replacement” is the exact same thing as “With Repetition.” It means that after you select an item from a pool, you physically put a duplicate item right back into the pool to replace it. Because the item was replaced, it is completely possible to randomly select that exact same item on your very next turn.
What is the “Stars and Bars” method?
The “Stars and Bars” theorem is the visual trick mathematicians use to understand the (n + r – 1)! formula. Imagine your selected items are “Stars”, and you separate your categories using “Bars”. If you are separating 5 categories, you only need 4 Bars. By adding your total Stars (r) and your total Bars (n-1) together, you find the total number of moving pieces in the probability equation.
Can my ‘r’ value be larger than my ‘n’ value?
Yes! In combinations with repetition, your number of selections (r) can absolutely be larger than your number of categories (n). For example, if you are buying a dozen donuts (r = 12), but the store only has 3 flavors to pick from (n = 3), the math still works perfectly because you are allowed to repeat flavors.