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I need your help. I’m really terrible at statistics, probability, permutations, and combinations. I’ve looked around a lot for the answer to this, but I haven’t seen anything just like it. I need the answer to this for some research that I’m doing.

Here’s the situation:

Imagine 60 bins or locations. Each bin is assigned one randomly-selected number, out of 256 possible numbers. It is possible that two or more bins can receive the same number.

My question is this: out of those 60 bins, on average, how many bins contain different numbers? Or, to put it another way, on average, how many different numbers are contained in the 60 bins?

Clearly, it’s possible that on some runs of this experiment, the 60 bins could all contain different numbers, so the count would be exactly 60. It’s also possible, although extremely unlikely, that all 60 bins could contain the same number, so the count would be 1.

From doing computer simulations of this situation, I get that the average count is between 53 and 54, which sounds reasonable. I just don’t know how to calculate that theoretical average count.

I also actually created simpler versions of this situation, with fewer bins and fewer numbers to assign to the bins. I then counted how many different numbers there were, and calculated the average. Here’s what I got:

For 3 bins and 4 numbers, the average count is 2.3125.

For 3 bins and 5 numbers, the average count is 2.44.

For 3 bins and 6 numbers, the average count is 2.527778.

For 3 bins and 7 numbers, the average count is 2.591837.

For 3 bins and 8 numbers, the average count is 2.640625.

So, a general formula for this situation should give these results.

I will be very grateful if you can tell me what the answer is for 60 bins and 256 numbers, and even more grateful if you can show me the formula for calculating this.

Thank you in advance!

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# Help! I need the expected value for a tricky situation!

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