5 different colored balls have matching 5 buckets. Each balls is randomly placed in a bucket. A bucket can only accommodate 1 ball.
What is the probability that:
a.) None of the balls match their buckets b.) Exactly 2 balls match their buckets
My approach is as follows:
a.) I just counted. Let's say I number the balls 1,2,3,4,5 with corresponding buckets, 1,2,3,4,5. I know that there are 4 different ways to get them all in the wrong buckets by just shifting the balls one bucket away. That is, (5,1,2,3,4) (4,5,1,2,3)(3,4,5,1,2) and (2,3,4,5,1). Then, another way to get them all wrong is to choose any 2, interchange their positions, then mix up the remaining three as well. There is only 1 way to mix up the remaining 3 once the first 2 are chosen so this is just 5C2 ways. This also results to the same outcome if instead I choose 3 first and jumble those before interchanging the other 2. So the probability is: $\frac{5C2+4}{5!}$
b.) For exactly 2 to match, we can choose any 2 from 5 and then mix up the other 3 so they won't match. This seems to exhaust all possible ways. So that's just 5C2, making the probability: $\frac{5C2}{5!}$