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

Enter the total number (n) and the number selected (r) and click the "Calculate Combinations" button to calculate and display the total number of combinations in which r unique items are selected from n.

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 total number of combinations is written as _nC_r, which represents the total number of combinations of numbers (r) selected from the total number (n).

For example, let's say you choose three letters from the four letters A, B, C, and D.

There are four ways to choose: ABC, ABD, ACD, and BCD.

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

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

To calculate the total number of combinations, you divide the total number of permutations by the number of identical combinations.

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

The total number of permutations is 4 x 3 x 2, or 24.

However, in permutations, there are six ways to arrange the numbers - "ABC, ACB, BAC, BCA, CAB, CBA" - but in combinations, there is only one way, as the order is not taken into consideration.

So the total number of possible permutations is 24, divided by 6, which gives us 4 combinations.

The only permutations that will result in the same combination are permutations of the numbers you choose, so if you choose three, you get 3!, which is 6 possibilities.

Therefore, the total number of combinations of selecting r items from n items can be calculated as _nP_rr!.

Also, since it is _nP_r = n!(n−r)!, it becomes _nP_rr! = n!(n−r)! × 1r! = n!r!(n−r)!.

Also, choosing r items from n is the same as not choosing the remaining (n-r) items from n.

Therefore, the total number of combinations of choosing r items is the same as the total number of combinations of choosing (n-r) items, and _nC_r and _nC_(n-r) are equal.