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Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that there is at least one pair that sums up to 7?

I interpreted the above as being equivalent to the following:

1 - (Probability that there is no pair that sums up to 7)

So if I were to consider just one pair of dice, then the probability that the pair adds up to 7 is 1/6, I think?

So Pr(one pair doesn't add up to 7) = 5/6.

But then I'm kind of stuck on how to proceed. Because there are lots of possible pairs amongst the n die, and some of these pairs overlap...for example, (die1, die2) is a pair, (die1, die3) is a pair, and so on. So I don't know how to account for these overlaps.

I tried breaking down the problem into a number of cases where there is no way for any pair to add up to 7:

(1) All of the dice show exactly one number.

(2) All of the dice show exactly two numbers which do not add up to 7 -- e.g. All the dice show either 3 or 6. Or all the dice show 2 or 4. And so on...

(3) All of the dice show exactly three numbers e.g. (1, 2, 4), no two of which can possibly add up to 7.

(4) If all of the dice show 4 or more numbers, then there MUST exist a pair that adds up to 7, so I don't consider any of these cases.

I suppose that I could add up the probabilities for all of the above cases, then I'd have the total probability that no two dice add up to 7? But then, how do I compute these?

In fact, is there a better / easier approach than the one I have thought up? Thanks.