Negative Binomial PMF from First Principles

Let $X$ be the number of independent Bernoulli($p$) trials needed to accumulate exactly $r$ successes.

(a) By considering the structure of the sequence of outcomes up to trial $X$, derive the PMF $P(X = k)$ for $k = r, r+1, r+2, \ldots$

(b) Verify that your PMF sums to 1 using the negative binomial series $(1-x)^{-r} = \sum_{j=0}^{\infty} \binom{r+j-1}{j} x^j$ for $|x| < 1$.

(c) Compute $P(X = 7)$ when $r = 3$ and $p = 1/2$.