If 6 balls are randomly placed in 3 bins, what is the probability that each bin will have 2 balls?

Question

I was working on probability problems and came across this. Can someone explain each step in detail how to get to the solution? Thanks so much.

Answer 1

There are $3^6$ equiprobable ways to place the $6$ balls in $3$ bins.

Now we must count in how many of these cases we end up with $3$ bins that contain $2$ balls.

First we select $2$ balls to be placed in the first bin, and for that there are $\binom62$ possibilities.

Then from the remaining $4$ we select $2$ balls te be placed in the second bin, and for that there are $\binom42$ possibilities.

Finally the remaining $2$ balls are placed in the third bin.

We found $\binom62\binom42$ possibilities, so that the corresponding probability on that is:$$3^{-6}\binom62\binom42$$