1972 USAMO Problems/Problem 3
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after selections (), the product of the numbers selected will be divisible by 10.
For the product to be divisible by 10, there must be a factor of 2 and a factor of 5 in there.
The probability that there is neither a factor of 2 nor 5 in there is . The probability that there is no 5 is , so the probability that there is a 2 but no 5 is . The probability that there is no 2 is , so the probability that there is a 5 but no 2 is . Thus the only possibility left is getting a 2 and a 5, and thus making the product divisible by 10. The probability of that is .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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