Problem
COM-B1-M02-P023 Three Numbers Not Adjacent
#23
★★★★☆ Level 4 of 5
How many permutations of \(1,2,\ldots,8\) have the property that no two of \(1,2,3\) are adjacent?
First arrange the other five numbers.
The numbers \(4,5,6,7,8\) can be arranged in \(5!\) ways. They create \(6\) gaps, including the ends. Choose \(3\) gaps for \(1,2,3\) and arrange these numbers in \(3!\) ways. Answer: \(5!\cdot20\cdot6=14400\).
Classic gap-method problem.