Question

Problem 4.

1. I’ve invited 6 friends over to my house, and each of them brings a cat with them. (It’s a cat party, obviously!) I decide to place the 6 distinct cats into 3 identical cat beds. In how many ways can I do this?

2. Prove the S(n, k) = S(n − 1, k − 1) + kS(n − 1, k).

3. Give an example of an onto function f : [6] → [3]. Give an example of a function which is not onto.

4. Prove that the number of onto function of f : [6] → [3] is 3!S(6, 3).

5. Use inclusion-exclusion to count the number of function f : [6] → [3] which are not onto.

6. Use the previous part to find a formula for S(6, 3).

7. At the end of the cat party, I hand back my guests’ cats at random. In how many ways can I do this so that no one gets their own cat back?

Answer 1

3-3) Nouw +eat b, has no pse image 8: Suent +hat b has no pre image uent tat ba has no pre image 2 ckims in which a as F muthor b mor b has any preim age i e. mly b3 bas pe trage unctuoS3 +n (Pn