Geometric or binomial distribution?

A monkey is sitting at a simplified keyboard that only includes the keys "a", "b", and "c". The monkey presses the keys at random. Let X be the number of keys pressed until the money has passed all the different keys at least once. For example, if the monkey typed "accaacbcaaac.." then X would equal 7 whereas if the money typed "cbaccaabbcab.." then X would equal 3.

a.) What is the probability $X\ge 10$?

b.) Prove that for an random variable Z taking values in the range {1,2,3,...}, $E(Z)=$ Summation from $i=1$ to infinity of $P(Z\ge i)$.

c.) What's the expected value of X?

First, is this a binomial distribution or a geometric distribution? I believe it is a binomial but my other friends says that it is geometric. As for the questions above, for a can I just do $1-P(X=9)\text{}\text{or}\text{}1-P(X9)$, but I don't know how I will calculate $X<9$, I would know how to calculate $P(X=9)$, I don't know how to do b or c.