Question

1. A small college is offering 5 math courses and 3 Computer Science courses. Each freshman must take 2 math courses and 1 Computer Science course. Show that if there are 91 freshmen, at least four of

Answer 1

If there are 91 freshmen at the small college and each freshman must take 2 math courses and 1 Computer Science course, then at least four of the math courses are being taken by more than one freshman.

Step-by-step explanation:

To demonstrate this, consider the worst-case scenario where each of the 91 freshmen takes exactly 2 distinct math courses, utilizing all 5 math courses. The total number of math course selections in this case is
`\(91 * 2 = 182\)`. However, since there are only 5 math courses available, this implies that at least 182 - 5) = 177 of the math courses are being taken by more than one freshman.

Now, since each freshman must also take 1 Computer Science course, there are a total of
`\(91 * 1 = 91\)`Computer Science course selections. With only 3 Computer Science courses available, it is evident that at least 91 - 3) = 88 of the Computer Science courses are being taken by more than one freshman.

Therefore, combining both cases, we can conclude that at least four of the courses (math or Computer Science) are being taken by more than one freshman. This ensures that there are at least four instances of a course being selected by multiple freshmen among the 91 freshmen.