[Verse 1]
Picture letters in envelopes, each one has a home
But chaos struck the mailroom, now none can find their own
Every single message lands somewhere it's not meant
This scrambled situation has a name that's quite elegant
[Chorus]
Derangement, derangement, nothing stays in place
Every element wanders to a different space
N factorial times the sum, alternating signs
One over K factorial, that's how we define
Approximately N factorial over E
Point three-six-seven-nine probability
[Verse 2]
Inclusion-exclusion helps us count what's gone astray
Let A-sub-i be permutations where i wants to stay
Union of these sets gives us the fixed-point crew
Subtract from total permutations, derangements peek through
[Chorus]
Derangement, derangement, nothing stays in place
Every element wanders to a different space
N factorial times the sum, alternating signs
One over K factorial, that's how we define
Approximately N factorial over E
Point three-six-seven-nine probability
[Bridge]
Remarkable convergence, even when N is small
One over E emerges, the most beautiful of all
Hat-check problem classic, guests receive wrong coats
Mathematical poetry in these scattered random notes
[Outro]
From zero up to N, the alternating dance
Minus one to the K power gives derangements their chance
No fixed points allowed here, complete rearrangement
This is the elegant chaos we call a derangement