Calculate the total number of combinations by selecting r items from n items, allowing repetition.

Enter the total number (n) and the number selected (r) and click the "Calculate Combinations" button. r items will be selected from the n items, allowing repetition, and the total number of combinations to be extracted will be calculated and displayed.

It also shows how to calculate the total number of combinations.

Please enter positive integers up to 10,000 for Total number and Selected number.

Combination means choosing and selecting some things from among different and distinguishable things.

The combination selected while allowing repetition is called Combination.

For example, let's say you have to choose two letters from three: A, B, and C.

You can choose the same thing multiple times, so there are six ways to choose: AA, AB, AC, BB, BC, and CC.

Unlike permutations, combinations do not consider order, so combinations that are the same when rearranged, such as AB and BA, are considered as one.

If we select two from A, B, and C, the combinations would look like this in a tree diagram.

When calculating the total number of Combinations with repetition, think about it using circles and dividers.

For example, suppose there are three letters, A, B, and C, and you have to choose four from them and think of combinations.

The four letters you choose will be represented as circles (〇), and since there are three different types of letters, you will use two dividers (｜) to arrange them.

Line up the circles and dividers and label the numbers ABC from left to right of the divider.

For example, if the combination is 〇〇｜〇｜〇, the combination will be AABC.Also, if the combination is 〇｜｜〇〇〇, the combination will be ACCC.

The total number of permutations of these circles and dividers is the total number of Combinations with repetition.

The total number of permutations of circles and dividers is 6!, and since there is no distinction between the circles and dividers, we divide it by (4! × 2!).

So the total number of combinations is 6!4!2! = 15.

Therefore, the total number of combinations obtained by selecting r items from n, allowing repetition, can be calculated as (r+n−1)!r!(n−1)!.