# Quick counting question

You have 10 pink balls and 15 purple balls. How many distinguishable arrangements are possible if two pink balls cannot be next to each other?

I know the answer is 8008 but I have no idea how to get this

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are arrangements of any size allowable? ie: 1 ball, 2ball, .... arrangements, or do they have to be a fixed size?

im pretty sure you have to use all 25 every time

I'm on the verge of a solution but the computer lab is closing so I'll have to wait till tomorrow to post , sorry. Hopefully someone else will help out till then.

They must mean arrangements of all possible sizes, since the total number of arrangements of size 25 is much much less than 8008. My method involves breaking the situation into cases based on how big the arrangement is. I will provide more tomorrow. Good luck

-MS

MathStudent is wrong your problem does not mean arrangements of all possible sizes.

The pink balls can not be touching each other so you need if you have 10 pink you need 9 purple.

Now you have 6 purple balls left to distribute and 11 places to put it.

This model is used in the Einstein solid to model the multiplicity of a solid with r atoms and N units of energy to distribute.
$$\Omega = \binom{N+r-1}{r-1}$$