5 Firms, 5 Cover Letters
Each personalized letter is randomly placed into an envelope. What is the probability that no letter goes to the correct firm?
Try a random assignment
Each row shows which letter lands in each firm's envelope.
Exact probability
11/30
≈ 36.67%
Total assignments5! = 120
Assignments with no matches!5 = 44
Probability44 / 120 = 11 / 30
A permutation with no fixed points is called a derangement.
Why are there 44 derangements?
Use inclusion–exclusion. Start with all 120 assignments, subtract those where at least one letter is correct, then add back cases counted twice, and so on.
All5!
120
Subtract 1 fixed− C(5,1)4!
−120
Add 2 fixed+ C(5,2)3!
+60
Subtract 3 fixed− C(5,3)2!
−20
Add 4 fixed+ C(5,4)1!
+5
Subtract 5 fixed− C(5,5)0!
−1
!5 = 120 − 120 + 60 − 20 + 5 − 1 = 44
P(no correct letter) = 44 / 120 = 11 / 30 ≈ 0.3667
Monte Carlo simulation
Run 10,000 random assignments and compare with the exact answer.
Expected result: about 36.67%.