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按学科分组摆放书籍

Arranging Books by Subject

专题
Statistics / 统计
难度
L4

题目详情

你有 5 本互不相同的数学书、4 本互不相同的物理书、3 本互不相同的化学书。

如果要求同一学科的书必须摆在一起，那么书架上一共有多少种摆放方式？

You have 5 distinct mathematics books, 4 distinct physics books, and 3 distinct chemistry books.
How many ways can you arrange these books on a shelf if all books of the same subject must be together?

解析

把每个学科的一组书看成一个“超级书块”，则有 3 个书块，先排列书块：3!3!。

每个书块内部再排列：数学 5!5!，物理 4!4!，化学 3!3!。

因此总数为

3!⋅5!⋅4!⋅3!=103,680.3!\cdot 5!\cdot 4!\cdot 3!=103{,}680.

Original Explanation

We treat each subject’s set of books as a single "super-book". This gives us 3 "super-books": one for mathematics, one for physics, and one for chemistry.

First, we arrange these 3 "super-books". This can be done in:

3!=6 ways3! = 6 \text{ ways}

Then, within each "super-book", we arrange the books of that subject:

  • Mathematics: 55 distinct books → 5!=1205! = 120 ways
  • Physics: 44 distinct books → 4!=244! = 24 ways
  • Chemistry: 33 distinct books → 3!=63! = 6 ways

Now, we multiply all these together:

3!×5!×4!×3!=6×120×24×6=103,6803! \times 5! \times 4! \times 3! = 6 \times 120 \times 24 \times 6 = 103,680

✅ Final Answer:

103, ⁣680\boxed{103,\!680}

So, there are 103, ⁣680103,\!680 ways to arrange the books on a shelf with all books of the same subject grouped together.