# Finding total number of cases in probability questions

A pool table has 7 holes through which 5 balls can drop. At each play, each ball is equally likely to go down any of 7 holes. In order to find the probability that each ball passes through distinct holes, I am confused whether the total number of cases is $$7^5$$ or $$5^7$$. I always get confused with the power while finding total possibilities.

It's useful to think of the "balls going through holes" as a process, and then keep track of the different possibilities so far. First, think of the first ball. It can go through $$7$$ holes. So we have $$7$$ cases for the first ball. Then the second ball comes, and it also has $$7$$ different holes that it can go through. Now we multiply these values and we have $$7\times 7 = 7^2$$ different cases.
• Is it wrong if I think that "a hole can hold any of 5 balls". Then it is 5 possibilities for hole 1. Like wise for the 7 holes wouldnt it be $5^7$.? – Harry Nov 27 '18 at 8:33
First you need to choose $$5$$ holes out of $$7$$ total in which balls drop. If the balls are indifferent, we have $$\binom{7}{5}$$ cases in total. Otherwise if all are distinct, the number of cases would become $$\binom{7}{5}5!$$ (we assume that the holes have a preserved arrangement)