Consider the colliding clowders.

Two clowders A and B, with n cats each, collide, forming three result clowders (C1;C2;C3)

as usual (where C1 only has cats from A, C2 only has cats from B, and C3 can have cats

from both of the original clowders).

(a) Assume each cat chooses randomly from among the result clowders that cat is

allowed to join. If the probability of any given cat choosing to go to C3 is p (the

same p for every cat), and the choices of all the cats are independent of each other,

what is the probability that C3 ends up containing exactly n cats? Write your

answer in terms of n and p. Use notation where appropriate.

(Hint: this problem is likely easier if you fix an ordering for the cats. First compute

the probability for a specific outcome, such as "the first n cats join C3 and the other

n cats do not", then multiply by the number of distinct permutations of that outcome,

being careful not to overcount duplicates.)

b) Repeat part (a), except the cats from B have probability r of going to

C3 (the cats from A still have probability p of going to C3).