Problem $1.(23$ points total$)$ In $2040,$ Boston University admits $2,000$ undergraduate computer science first$-$years due to an explosion in the major$$s popularity. To accommodate these CS first$-$ years, each of the $800$ rooms in Warren Towers has been expanded into a luxurious quad to hold four students each. All of the students in this class cohort are forced to be housed in Warren Towers.

To align interests between students, BU decides to group Warren Towers$$ rooms by major. This means that each CS first$-$year is placed in a Warren quad with three other CS first$-$years as their roommates; a room has either four CS first$-$years or no CS first$-$years at all.

Please briefly explain your answers in each part of this problem; one to two English sentences is fine. Numerical answers with no explanation will be worth nothing. Answers do not need to be calculated out, and can contain exponents, factorials, C$($n$,$ r$),$ and P $($n$,$ r$).$

a$)(5$ points$)$ Let a room that houses four CS first$-$years be called a CS room. How many CS rooms would BU need to accommodate the $2,000$ CS first$-$years? How many ways can BU distribute the CS rooms among the $800$ total rooms in Warren Towers?

b$)(5$ points$)$ How many ways can the CS first$-$years be distributed among the CS rooms?

c$)(6$ points$)$ To ensure a smooth on$-$boarding of students, the CS department instates a mandatory CS$-$specific orientation. $1,556$ of the $2,000$ students attend. Show that at least a group of four students from the same CS room attended this orientation.

d$)(7$ points$)$ To adjust to the rise in CS majors, CS$131$ chooses to raise its number of seats to $600.560$ first$-$years enroll in the course. What is the minimum number of CS rooms that have more than one of its residents taking CS$131?$