Question

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 the 7 courses must be astatistics course. Suppose that 5 of the 20 courses are statistics courses.(a) How many choices are there for which 7 courses to take?(b) Explain intuitively why the answer to (a) is not

Answer 1

Complete answer:

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 the 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.

(a) How many choices are there for which 7 courses to take?

(b) Explain intuitively why the answer to (a) is not $\binom{5}{1}\binom{19}{6}$

Answer:

a) 71085 choices

b) See below

Step-by-step explanation:

a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's: $\binom{20}{7}$, but now we should subtract from that all the possibilities when none of the courses chose are a statistic course, that's is $\binom{15}{7}$ because 15 courses are not statistics and 7 are the ways to arrange them. So finally, the choices for which 7 courses to take with the constraint that at least 1 of the 7 courses must be a statistics course are:

$\binom{20}{7}-\binom{15}{7}=71085$

b) It's important to note that the constraint at least 1 of the 7 courses must be a statistics course make the possible events dependent, we can not only fix an statistic course and choose the others willingly ( that is what $\binom{5}{1}\binom{19}{6}$ means) because the selection of one course affect the other choices.