This puzzle is challenging, but it should be fun.

There are a lot of interesting math problems that can be built around finding probabilities. Suppose we are doing a random experiment (such as tossing a coin or throwing a die) where all outcomes are equally likely. We can turn the experiment into a game by deciding which outcomes will be "wins" and which outcomes will be "losses". (For example, we might decide that we will "win" a dice throwing game if the number on the die turns out to be 1 or 5. The outcomes 2, 3, 4, and 6 will mean we "lose" the game.) The probability of winning is then just a simple fraction:

[the number of "winning" outcomes] / [the total number of all outcomes]

(So in our simple dice throwing game, there is a probability of 2/6 we will win, since there are two winning outcomes and 6 total outcomes.)

Now here is a harder example. A bin contains 25 balls: 10 red, 8 yellow, and 7 blue. We draw three balls at random (without looking!) from the bin, and we will say that we "win" if our three balls represent exactly two colors. (That is, we "win" if we draw two balls of one color and another ball of a different color.)

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What is the probability of winning this particular game?