AMC8 2018

AMC8 2018 · Q16

AMC8 2018 · Q16. It mainly tests Basic counting (rules of product/sum), Permutations.

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

张教授有九本不同的语言书籍排放在书架上：两本阿拉伯语，三本德语，四本西班牙语。保持阿拉伯语书籍在一起且西班牙语书籍在一起，将这九本书排放在书架上的方法有多少种？

(A) 1440 1440

(B) 2880 2880

(D) 182,440 182,440

(E) 362,880 362,880

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): There are two ways to arrange the two Arabic books among themselves and there are $4\cdot3\cdot2\cdot1=24$ ways to arrange the four Spanish books among themselves. If the two Arabic books are considered as a unit and the four Spanish books are considered as a unit, then, including the three German books, there are five objects to arrange on the shelf. These five objects can be arranged on the shelf in $5\cdot4\cdot3\cdot2\cdot1=120$ ways. So altogether the books can be arranged in $2\cdot24\cdot120=5760$ ways.

答案（C）：两本阿拉伯书在内部可以有 2 种排列方式，四本西班牙书在内部有 $4\cdot3\cdot2\cdot1=24$ 种排列方式。若把两本阿拉伯书看作一个整体、把四本西班牙书看作一个整体，再加上三本德语书，则书架上相当于有 5 个“对象”需要排列。这 5 个对象在书架上的排列方式有 $5\cdot4\cdot3\cdot2\cdot1=120$ 种。因此总的排列方式为 $2\cdot24\cdot120=5760$ 种。

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