Problem
COM-B2-M02-P008 Three Forbidden Positions
#8
★★★☆☆ Level 3 of 5
How many permutations of \(1,\ldots,8\) place \(1,2,3\) away from their own positions?
There are only three restrictions; the other numbers may be fixed.
Without restrictions, \(8!\). If chosen \(i\) among \(1,2,3\) are fixed, the remaining \(8-i\) elements are permuted freely. The answer is \(8!-\binom31 7!+\binom32 6!-\binom33 5!=27240\).
This distinguishes partial restrictions from full derangements.