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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.

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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.

Can you see how to continue from here?

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  • $\begingroup$ 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 $5^7$.? $\endgroup$ – user46697 Nov 27 '18 at 8:33
  • $\begingroup$ With that thinking, the process would look like this: First, the first hole chooses one of 5 balls. Then the second hole again chooses one out of 5 balls ... So this approach assumes that every hole takes exactly one ball, and somehow the balls are also able to duplicate! For example, it would be possible that ball #1 goes to all of the holes. Clearly, this is wrong. $\endgroup$ – Matti P. Nov 27 '18 at 8:55
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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)

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