# Determining how many combinations there are when every item has a pair it can't exist with.

If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want to choose 25 balls, but no person can have both of their balls in the choice?

I've taken some specifics out of this example, but its to help me decide if I should programmatically brute force an optimization problem, or use some algorithm to get to the same thing.

I know that for example, $\binom{50}{25}$ is the magnitude of $10^{14}$, which would take too long. I know that my problem should have a number less than that, however I don't know how to work that out!

• There are $2$ choices for which ball we will take from Person 1. For each choice, there are $2$ choices for which ball we will take from Person 2. And so on. – André Nicolas May 26 '15 at 21:06

Since , $1$ ball is to be taken from each person and each person has $2$ choices: $$2^{25}$$