Take seven courses out of 20 with requirement 
To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.
From Introduction to Probability, Blitzstein, Hwang

Why is ${5 \choose 1}{19 \choose 6}$ not the correct answer?

The animated video from the stat110x course gives a clear picture of why Vandermonde's Identity instead of $\binom{5}{1} · \binom{19}{6}$ is used. 
Vandermonde's Identity: 
$ \binom{m+n}{r} = \sum_{k=0}^r \binom{m}{k} \binom{n}{r-k} $
Using the video as reference for the stat class question, $\binom{m}{k}$ represents the stat classes and $\binom{n}{r-k}$ represents the none stat classes. For every stat class chosen, the number of $r$ choices reduces, hence, r-k from when choosing the none stat classes.
$ \binom{5+15}{7} - \binom{5}{0}\binom{15}{7} = \binom{20}{7} - \binom{15}{7}$
We minus $\binom{5}{0}\binom{15}{7}$ because this represents the permutations of choosing 0 stat class and 7 none stat classes.
