Problem
COM-B2-M02-P012 Two Pairs of Restrictions
#12
★★★☆☆ Level 3 of 5
How many permutations of \(1,\ldots,6\) place \(1\) away from position \(1\), \(2\) away from position \(2\), \(3\) away from position \(3\), and \(4\) away from position \(4\)?
This is a partial derangement with four restrictions.
By inclusion-exclusion:
\[6!-\binom41 5!+\binom42 4!-\binom43 3!+\binom44 2!=720-480+144-24+2=362.\]
The numbers \(5\) and \(6\) may remain fixed.