How to calculate possible combinations when 1 object is always selected?

News & General Discussion
1

There is a formula to calculate possible combinations - nCr = n! / r! * (n - r).
But what happens when the last or any 1 item is always selected.

Example:
A number objects to select in any order, with a certain amount selected at any time, but with 1 requirement. The last object is always selected no matter what.

I am trying to find the number of combinations in 3 different scenarios.
10 total objects to select in any order with 3 objects selected at time + the 10th object that is always selected. Total of 4.
12 total objects to select in any order with 5 objects selected at time + the 12th object that is always selected. Total of 6.
14 total objects to select in any order with 7 objects selected at time + the 14th object that is always selected. Total of 8.

I am making a game and I want to know the total combinations I will have with the scenarios above.

Thanks for any help.

2

You could loop the structures and count the iterations/ spit out index values to console.

3

After testing some combinations, if a number or object cannot be changed it is not included in the combination.
My 3 scenarios would be:
9 total objects to select in any order with 3 selected at any time. That would be 84 combinations.
11 total objects to select in any order with 5 selected at any time. That would be 462 combinations.
13 total objects to select in any order with 7 selected at any time. That would be 1760 combinations.

The total of all the combinations would be 2306 using nCr = n! / r! * (n - r).

4

I’m not a math expert but I’m pretty sure that is the right way to do things. The fixed items would not change the number of combinations, you basically just append them onto the combinations of the items that are not fixed.