Q:

Can someone help me understand this pmf question? 

A lot, consisting of 100 fuses, is inspected by the following procedure. 
Five of these fuses are chosen at random and tested; if all five “blow” at the correct amperage, the lot is accepted.  If, in fact, there are 20 defective fuses in the lot, the probability of accepting the lot is, under appropriate assumptions,
$\frac{80\choose5}{100\choose5}$
More generally, let the random variable X be the number of defective fuses among
the five that are inspected.  The pmf of X is given by

I have no idea what the $20\choose x$ and $80\choose 5-x$ mean at all.  What's it supposed to signify, and why is that the probability? 

A:

This is an example of a hypergeometric random variable. 
First, for the specific question asked.  If there are
80 good fuses and 20 bad fuses, what is the probability
of choosing 5 fuses at random without replacement, all
of which are good. 
The number of ways of choosing 5 fuses from among the 80 is ${80 \choose 5} = \frac{80!}{5!\cdot 75!}  =  24\,040\,016.$
The number of ways of choosing 5 fuses from among among all 100 fuses is ${100 \choose 5} = 75\,287\,520.$
So the probability of getting 5 good fuses when drawing 5 fuses from the lot with 80 good and 20 bad fuses is the ratio  $0.3193094.$
Second, in the displayed equation, Let $X$ be the number of bad fuses chosen.  Then the probability of getting $x=0$ bad fuses is the same as the probability of getting 5 good ones. 
$$p_X(0) = \frac{{20 \choose 5}{80\choose 0}}{100 \choose 5}.$$
This is the same as in the previous part because
${80 \choose 0} = 1$
The probability of getting exactly $x=1$ bad fuses is
$$p_X(1) = \frac{{20 \choose 4}{80\choose 1}}{100 \choose 5}.$$
And so on. 
Addendum: In R, phyper with appropriate arguments the PDF of a particular hypergeometric distribution.  We can make a bar graph of the random variable $X$ above, which counts bad fuses out of five from the specified lot. 
x = 0:5;  pdf = dhyper(x, 20,80, 5)
plot(x, pdf, type="h", lwd=2, 
     main="Hypergeometric Dist'n of Bad Fuses")
  abline(h=0, col="green2")

