AMC 12A 2025 (Problem 13)
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?
,
From each set, we should remove at least one element.
From each set, we should remove at least one element.
Consider the subset .
By removing and , there are no five consecutive integers.
By removing and , there are no five consecutive integers.
From Hints 1 and 2,
Let the two numbers that are removed be and , where .
To avoid having five consecutive integers:
To avoid having five consecutive integers:
Possible pairs are:
From Hints 4 and 7:
(D)