Problem:
Let . Let be the greatest integer such that there exists a subset of with elements that does not contain five consecutive integers. Suppose integers are chosen at random from without replacement. What is the probability that the chosen elements do not include five consecutive integers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If we skip only one element from , then there will be at least elements greater than the skipped one or elements less.
Therefore we must skip elements and it is easy to see that this is possible e.g. skipping and .
Now, we must compute the number of ways to delete elements while ensuring no numbers are consecutive.
There are only ways to do this - skipping or for a probability of .
The problems on this page are the property of the MAA's American Mathematics Competitions