1.1.8 problem 10

(a)

Case 1: Student takes exactly one statistics course.

There are $5$ choices for the statistics course. There are $\left(\frac{15}{6}\right)$ choices of selecting $6$ non-statistics courses.

Case 2: Student takes exactly two statistics courses.

There are $\left(\frac{5}{2}\right)$ choices for the two statistics course. There are $\left(\frac{15}{5}\right)$ choices of selecting $5$ non-statistics courses.

Case 3: Student takes exactly three statistics courses.

There are $\left(\frac{5}{3}\right)$ choices for the three statistics course. There are $\left(\frac{15}{4}\right)$ choices of selecting $4$ non-statistics courses.

Case 4: Student takes exactly four statistics courses.

There are $\left(\frac{5}{4}\right)$ choices for the four statistics course. There are $\left(\frac{15}{3}\right)$ choices of selecting $3$ non-statistics courses.

Case 5: Student takes all the statistics courses.

There are $\left(\frac{15}{2}\right)$ choices of selecting $2$ non-statistics courses.

So the total number of choices is

$$\left(\frac{5}{1}\right)\times \left(\frac{15}{6}\right)+\left(\frac{5}{2}\right)\times \left(\frac{15}{5}\right)+\left(\frac{5}{3}\right)\times \left(\frac{15}{4}\right)+\left(\frac{5}{4}\right)\times \left(\frac{15}{3}\right)+\left(\frac{5}{5}\right)\times \left(\frac{15}{2}\right)$$

An Alternative Approach

There are $\left(\frac{20}{7}\right)$ choices of selecting 7 courses which is the maximum number of choices if there were no restriction as choosing at least one statistics course.

$\left(\frac{15}{7}\right)$ is the number of choices without any statistics course.

So the total number of choices with at least one statistics course is

$$\left(\frac{20}{7}\right)-\left(\frac{15}{7}\right)$$

(b)

It is true that there are $\left(\frac{5}{1}\right)$ ways to select a statistics course, and $\left(\frac{19}{6}\right)$ ways to select $6$ more courses from the remaining $19$ courses, but this procedure results in overcounting.

For example, consider the following two choices.

(a)

STAT110, STAT134, History 124, English 101, Calculus 102, Physics 101, Art 121

(b)

STAT134, STAT110, History 124, English 101, Calculus 102, Physics 101, Art 121

Notice that both are selections of the same $7$ courses.