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?

There are $\binom{20}{7}$ possibilities, but $\binom{15}{7}$ of them are not ok, hence the answer is $\binom{20}{7}-\binom{15}{7}$.
