Math CentralQuandaries & Queries


Question from Antwan, a student:

I have 40 numbers........number 1-40.
I want to know how many times i can chose 10 of those numbers without picking the same exact sequence twice if its even possible?


Anything is possible, even things that are very unlikely.

It is possible for me to flip a coin and for it to come up heads 50 times in a row, but that doesn't happen often. If it happened, we'd certainly think the situation had a lot more to do with a trick, deception, or a problem with the coin than just luck.

The same is true of what you describe.

Since you used the word "sequence" I assume the order the numbers are selected is important.

So let us calculate just how unlikely your situation is.

I have to make certain assumptions because I don't have all the necessary details. I will assume that you pick 10 numbers without replacement. That means that none of the 10 numbers can match each other.

Also, the second occurrence of this selection occurs with a fresh set of all 40 numbers.

One or two other possibilities occurs to me that may be what you are trying to describe, but I think this is what you mean.

So start by getting the first 10 numbers and write them down.

Now start again with all 40 numbers and pick one randomly. It must match the first number you chose earlier. Since you selected from a pool of 40 possible numbers, the chance that you chose the "same" one is 1 in 40 (written as a fraction: 1/40).

Pick the second number randomly. It is drawn from a pool of 39 numbers (remember that the first number has been removed). Again, there is just one chance that you chose the "same" number as you chose in the first set of 10. So the chance here is 1/39.

The rest would be 1/38, 1/37, ... , 1/31.

To find the overall likelihood of the problem, we just multiply all these fractions together.

This is (1/40) x (1/39) x (1/38) x ... x (1/31), which my calculator says is 1 in 3,075,990,524,006,400.

This is three times harder to do than flipping a coin 50 times and getting heads each time!

Stephen La Rocque.

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