Repetition means doing one thing multiple times. In combinatorics, repetition refers to drawing with replacement. This means that the drawn object is returned after each drawing to the set of objects from which is drawn. Therefore, easily more objects can be drawn than are available, because each element can be drawn multiple times, i.e., repeatedly. Each drawing therefore proceeds under the same conditions. For illustration, the urn model is usually chosen: an urn with various balls drawn from it. In fact, the type of objects does not matter, but this calculation assumes that all objects are clearly distinguishable from one another.
| ( | n+k-1 | ) | = | (n+k-1)! |
| k | k!(n-1)! |
Number of possibilities when drawing with replacement. k drawings are made from n elements. Drawing with replacement offers more possibilities than drawing without replacement. The latter is calculated using the binomial coefficient. The calculation with replacement is also performed using the binomial coefficient, but in this case, k is added to the top row and 1 is subtracted. Therefore, this method also works with few objects and many drawings, since the number of objects and the number of drawings are added together.
Enter natural numbers (positive integers) for n and k and then press Calculate.
An example: you have four flavors of ice cream and want six different scoops, so n=4 and k=6. So there are 84 different possibilities. Of course, a scoop of ice cream isn't replaced, but it is assumed that there is an unlimited amount of ice cream available. If you draw a scoop of vanilla ice cream, for example, there will still be enough vanilla ice cream left for more scoops. While a scoop was removed, in the model, no ice cream was effectively removed. This amounts to the same thing as putting the scoop back after drawing.
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