A math problem:

There are $\displaystyle 42$ students who are to share $\displaystyle 12$ computers. Each student uses exactly $\displaystyle 1$ computer and no computer is used by more than $\displaystyle 6$ students. Show that at least $\displaystyle 5$ computers are used by $\displaystyle 3$ or more students.

One kind of proof using an argument by contradiction:

Suppose not. Suppose that $\displaystyle 4$ or fewer computers are used by $\displaystyle 3$ or more students.[A contradiction will be derived.]Then $\displaystyle 8$ or more computers are used by $\displaystyle 2$ or fewer students.

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(End of first proof)

My question 1:

How did the author deduce that "Then $\displaystyle 8$ or more computers are used by $\displaystyle 2$ or fewer students"?

Another kind of proof using a direct argument:

Let $\displaystyle k$ be the number of computers used by $\displaystyle 3$ or more students.[We must show that $\displaystyle k \geq 5$] Because each computer is used by at most $\displaystyle 6$ students, these computers are used by at most $\displaystyle 6k$ students(by the contrapositive form of the generalized pigeonhole principle). The remaining $\displaystyle 12 - k$ computers are each used by at most $\displaystyle 2$ students.

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(End of this proof)

Another question 2:

Why's that "The remaining $\displaystyle 12 - k$ computers are each used by at most $\displaystyle 2$ students."?