If two dice are thrown 5 times what is the probability that sum of the numbers coming up will be 10 in exactly 5 out of these 5 trials.

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I know there can be 3 possible pairs, so we have 1/12 possibilities to toss one of them. But I'm not sure how to prove it for 5 out of 5 times.

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You want a ten on all five tosses, which are independent, so the probabilities multiply. The answer is $\frac1{12^5}$.