2018 AMC 10A Problem 4

Question

How many ways can a student schedule

3

mathematics courses — algebra, geometry, and number theory — in a

6

-period day if no two mathematics courses can be taken in consecutive periods?

(What courses the student takes during the other

3

periods is of no concern here.)

3

6

12

18

24

First choose the three nonconsecutive periods

After the periods are chosen, arrange the three named courses

The

3

classes can occupy the following periods:

(1, 3, 5),

(1, 3, 6),

(1, 4, 6),

(2, 4, 6).

This means that there are

4

ways to choose which periods the mathematics courses occur.

For each configuration, there are

3!

ways to determine the order of the courses, for a total of

6 \cdot 4 = 24

schedules.

Thus, E is the correct answer.

Problem 4 in other years