AMC12 2018 A

AMC12 2018 A · Q3

AMC12 2018 A · Q3. It mainly tests Basic counting (rules of product/sum), Casework.

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.)

学生如何在6节课的一天中安排3门数学课——代数、几何和数论，使得没有两门数学课在连续节次？（其他3节课上什么课无关。）有多少种方式？

(A) 3 3

(B) 6 6

(D) 18 18

(E) 24 24

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): There are 4 choices for the periods in which the mathematics courses can be taken: periods 1, 3, 5; periods 1, 3, 6; periods 1, 4, 6; and periods 2, 4, 6. Each choice of periods allows $3! = 6$ ways to order the 3 mathematics courses. Therefore there are $4 \cdot 6 = 24$ ways of arranging a schedule.

答案（E）：数学课程可以安排的时段有 4 种选择：第 1、3、5 节；第 1、3、6 节；第 1、4、6 节；以及第 2、4、6 节。每一种时段选择都允许用 $3! = 6$ 种方式对 3 门数学课进行排序。因此共有 $4 \cdot 6 = 24$ 种安排课表的方法。

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