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:
The case 2 answer kind of bothers me since we flip flop between permutations and combinations.
Thank you for your help.