# How many of these permutations leave no number fixed?

I am not exactly sure how to approach this problem. I know that there exists only one mapping in which all numbers are fixed, but I don't how this translates into leaving no number fixed. Below are the rest of the problems I have to solve, but I am confident that if I can solve the first, then it will help me understand the rest.

We know that there are n! different permutations of the set {1, 2,...,n}.

(a) How many of these permutations leave no number fixed?

(b) How many of these permutations leave at least one number fixed?

(c) How many of these permutations leave exactly one number fixed?

(d) How many of these permutations leave at least two numbers fixed?

For each part of this problem, give a formula or algorithm that can be used to compute the answer for an arbitrary value of n, and then compute the value for n = 10 and n = 26.