Handing out distinct presents

The problem is written as follows:

Each of 10 employees brings one (distinct) present to an office party. Each present is given to a randomly selected employee by Santa (an employee can get more than 1 present). In how many ways can at least two employees get no presents?

The total number of ways to give out presents without restriction is:

I am going to count this using the complement, i.e 0 people recieve no presents or 1 person recieves no presents.

Case 1: 0 people receive no presents

Each person must get at least 1 present, so everyone gets exactly 1 present. Number of ways:

Case 2: 1 person recieves no presents

First, we pick the person who gets no presents:

Some other person must get two presents:

Pick 2 of the 10 presents to give that person:

Hand out the remaining 8 presents (one to each of the remaining 8):

Answer for this case:

Overall answer:

The case 2 answer kind of bothers me since we flip flop between permutations and combinations.

Thank you for your help.