I'm confused about how to approach this problem. If someone could give me a start, I would really appreciate it!

Suppose n men throw their coats into a closet and later pick them up blindly - one coat per man, uniform probability that any particular coat will go to any particular man.

Let B_n be the number of ways of permuting n coats so that every coat gets moved (i.e. B_2 = 1 because there is 1 way of switching 2 coats so that each coat gets moved).

Let C_n be the number of permutations (i.e. C_2 = 2 because there are 2 ways of arranging 2 objects. In general, C_n = n!).

B_n divided by C_n equals D_n.

Prove that D_n - D_{n-1} = -1/n (D_{n-1} - D_{n-2}).