Combinatorics & Probability

Question

A rumor is spread randomly among a group of 10 people by successively having one person call someone, who calls someone, and so on. A person can pass the rumor on to anyone except the individual who just called.

(a) By how many different paths can a rumor travel through the group in three calls?

(b) By how many different paths can a rumor travel through the group in $N$ calls? 

(c) What is the probability that if $A$ starts the rumor, then $A$ receives the third calls?

Comments

Since the group size is fixed and the rumor can be started by anyone from the group, any individual of the group who "starts" the rumor can do so by sharing it with any of the 9 remaining members of the group. For obvious reasons, I've assumed that a person cannot call himself.

The next person, who was the first recipient of the rumor, cannot pass it to the person who he/she just received the rumor from, or to himself. That leaves him with 8 choices.

This pattern than repeats and every following person will have 8 choices

a) Three calls: 9 * 8 * 8

b) N calls: 9 * (8)^(N-1)

c)

Total possible sequences of three calls = 9 * 8 * 8

Favorable sequences (where third call is recieved by A) = 9 * 8 * 1 ( The fact that A started the rumor is irrelevant, because, even in that case, A will have 9 choices, the next person 8, but the next person only 1 i.e.. A)

Probability = favorable / total = 9 * 8 * 1 / 9 * 8 * 8 = 1/8.