Some five courses are offered on campus in a semester. The group of students who

Question

Some five courses are offered on campus in a semester. The group of students who take at least one of these courses consists of 155 students. There are 80 students registered in each course. For any two among these courses, there are precisely 40 students who take each one. For any three of these courses, there are precisely 20 students who take each one. For any four among these courses, there are precisely 10 students who take each one. How many students in this group take every one among the five courses in that memorable semester?

Explanation / Answer

Let A, B, C, D, E denotes five course.

Here we need to find P(A B C D E)

Then we have the formula as,

P(A U B U C U D U E) = P(A)+P(B)+P(C)+P(D)+P(E)-P(A B)-P(A C)-P(A D)-P(A E)-P(B C)-P(B D)-P(B E)-P(C D)-P(C E)-P(D E)+P(A B C)+P(A B D)+P(A B E)+P(A C D)+P(A C E)+P(A D E)+P(B C D)+P(B C E)+P(B D E)+(C D E)-P(A B C D)-P(A B C E)-P(A B C D)-P(A C D E)-P(B C D E)+P(A B C D E)

There are 80 students registered in each course so we have,

P(A)=P(B)=P(C)=P(D)=P(E) = 80

For any two among these courses, there are precisely 40 students who take each one so we have,

P(A B) = P(A C) =P(A D) = P(A E) =P(B C) = P(B D) = P(B E) =P(C D) =P(C E) =P(D E) = 40

For any three of these courses, there are precisely 20 students who take each one, so we have,

P(A B C)=P(A B D)=P(A B E)=P(A C D)=P(A C E)=P(A D E)= P(B C D)=P(B C E)+P(B D E)+P(C D E) = 20

For any four among these courses, there are precisely 10 students who take each one, so we have,

P(A B C D)=P(A B C E)=P(A B C D)=P(A C D E)=P(B C D E) = 10

As the group of students who take at least one of these courses consists of 155 students.

P(A U B U C U D U E) = 155

We substitute these values in the formula we get,

155 = 80+80+80+80+80-40-40-40-40-40-40-40-40-40-40+20+20+20+20+20+20+20+20+20+20-10-10-10-10-10+ P(A B C D E)

155 = 400 – 400 + 200 – 50 + P(A B C D E)

155 + 50 = 200 + P(A B C D E)

P(A B C D E) = 205 – 200 = 5

Hence, for any five among these courses, there are precisely 5 students who take each one.

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