Combinations with mutually exclusive objects

[Ryoukomaru]

How do u calculate the the total number of combinations, given that you have n number of object and you will choose r of the objects, but x of these objects are mutually exclusive. Let x=2 for your explanations.

I kinda have an idea on how to do this, but i can't frecall an formula for the calculations.

Total number of permutations would be $$\frac{n!}{(n-r)!r!}$$ And now i have to subtract from this, the number of combinations with one of the mutually exclusive events given that the other has happened.

P.S. Maybe i should post a question to help you understand better ? ..., as my linguistic skills are not top-notch.

 

[Bartholomew]

If the objects are mutually exclusive that means you can only select 1 of them. So you choose from n - x + 1 objects, and multiply it by x ways to choose from the x objects.

 

[noelhustler]

I don't think that's quite right. It ignores the combinations that don't have one of the x objects.

 

[noelhustler]

A solution for this problem is:

$$\binom{n}{k}-\sum_{i=2}^{x}\left[\binom{x}{i}\times\binom{n-x}{k-i}\right]$$

I'm sure there's a more elegant formulation, but this one works.

Of course, for x=2, this simplifies to:

$$\binom{n}{k}-\binom{n-2}{k-2}$$