Difference between revisions of "1972 USAMO Problems/Problem 3"
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Revision as of 14:33, 30 December 2008
Problem
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.
Solution
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 not a factor of 2 or 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 
.
See also
| 1972 USAMO (Problems • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
