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 given answers are all correct. Nevertheless, there was one argument which made it very clear (to me) that 
$$
{5 \choose 1} {19 \choose 6}
$$
is not the correct answer. It goes as follows:
Question: Suppose there are 20 different courses and five of those focus on statistics. You have to choose one "major" and six side courses. How many different combinations are possible, if the "major" must focus on statistics?  
Answer: We have to choose one out of the five statistics courses and six out of the remaining 19 courses. Thus, the answer is 
$
{5 \choose 1} {19 \choose 6}
$. 
However, in the original question asked by Blitzstein, there is no reference to a "major" course. Thus, the original question (without the major course) has fewer arrangements. 
