Problem
COM-B2-M02-P014 Exactly \(k\) Fixed Points
#14
★★★★☆ Level 4 of 5
Prove that the number of permutations of \(n\) elements with exactly \(k\) fixed points is \(\binom nkD_{n-k}\).
First choose which points will be fixed.
Choose the \(k\) elements that will be fixed: \(\binom nk\) ways. The remaining \(n-k\) elements must have no fixed points among their positions; otherwise there would be more than \(k\) fixed points. This can be done in \(D_{n-k}\) ways. The result is \(\binom nkD_{n-k}\).
This formula often appears in olympiad problems with “exactly”.