Cosmin has three ping pong balls, and ten cups in front of him. If he throws the ping pong balls into the cups, what is the probability that exactly two balls land in the same cup? (A ping pong ball has an equal probability of landing in any cup.)
Let the balls be Red Blue and Yellow. My method requires that the ball be thought of as different
the pairs are RB, RY and BY so there are 3 ways to pair them.
Number of ways the balls can go into the cups is 10*10*10 = 1000
Number of ways to get Blue and Red together and Yellow somewhere else = 10*1*9=90
there are 3 possible pairings so that is 90*3=270
So P(exactly 2 in the same cup) = 270/1000 = 27/100
Another way to look at it is by subtraction:
the number of ways they can all be in different cups is 10*9*8 = 720
The number of ways they can all be in the same cup is 10*1*1 =10
so the number of ways that 2 will be in a cup is 1000-720-10 = 270
the answer is the same so that is conclusive enough for me.