5.4 due on 11/9

Difficult
The theorem for inclusion-exclusion inequalities makes sense, but I don't understand the proof. I don't know how or why there are combinations and from then on, I can't follow the rest of the proof. For the equation of a geometric random variable, it may be a small detail, but I thought the equation raised (1-p) to the (k-1) power instead of the k power. Unless X is the random variable representing the number of tails before the first head, instead of the total number of tosses. In part (b) of Example 13, I'm not sure how we came up with the probability of z tails. How did we get (z+n-1) C (n-1) ?
Reflective
The examples with the binomial random variable weren't as difficult as the rest of the section since the binomial RV was introduced as a series of Bernoulli trials. Even though there's a theorem stating any discrete RV can be written as a linear combination of Bernoulli trials, the binomial RV is the only one I can clearly visualize. I wonder how it is possible to represent a Poisson RV as a series of Bernoulli trials?