Let $D_n$ be the derangement count, prove:

• $D_n =n\cdot D_{n−1} +(−1)^n$
• $D_n = (n−1)\cdot (D_{n−2} +D_{n−1})$

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What probability that a reader guesses at random will match three

• all correctly?
• all incorrectly?