Math Problem Statement

In how many ways can I arrange $3$ different math books and $4$ different history books on my bookshelf, if at least two of the history textbooks are next to each other?

Solution

To solve this problem, we'll break it down into two steps:

Step 1: Total number of ways to arrange all the books.

There are 3 math books and 4 history books, so we have 77 books in total to arrange. The total number of ways to arrange them, without any restrictions, is:

7!=7×6×5×4×3×2×1=5040 ways.7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \text{ ways.}

Step 2: Number of ways where no two history books are together.

Now, let's calculate the number of ways in which no two history books are adjacent. We can think of this as arranging the math books first, and then placing the history books in the gaps between the math books.

  • There are 3 math books, so when we arrange them, there are 4 gaps (before the first math book, between the math books, and after the last math book) where the history books can be placed.

  • We want to place all 4 history books in these 4 gaps, but since there are only 4 gaps and 4 history books, each gap must contain exactly one history book. This can be done in:

4!=4×3×2×1=24 ways.4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways.}

  • Next, arrange the math books, which can be done in:

3!=3×2×1=6 ways.3! = 3 \times 2 \times 1 = 6 \text{ ways.}

Thus, the number of ways to arrange the books such that no two history books are adjacent is:

6×24=144 ways.6 \times 24 = 144 \text{ ways.}

Step 3: Number of ways where at least two history books are together.

Now, we subtract the number of ways where no two history books are adjacent from the total number of arrangements:

5040144=4896 ways.5040 - 144 = 4896 \text{ ways.}

Thus, the number of ways to arrange the books such that at least two history books are next to each other is 48964896.

Final Answer:

There are 4896\boxed{4896} ways to arrange the books such that at least two of the history textbooks are next to each other.


Would you like more details or have any questions on this? Here are 5 questions you can explore next:

  1. How does the arrangement change if all the math books must be together?
  2. What if instead of 4, there were 5 history books to arrange?
  3. How many ways can you arrange the books if all history books must be together?
  4. Can you generalize this problem for mm math books and hh history books?
  5. What if the order of the books matters only for the math books?

Tip: Always consider the restrictions in combinatorics problems carefully; breaking them down into smaller, manageable steps can simplify the solution.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations
Arrangement with Restrictions

Formulas

Total arrangements: n! = n × (n-1) × (n-2) × ... × 1
No adjacent arrangement: m! × (gaps)!
At least two together: Total arrangements - No adjacent

Theorems

Factorial Theorem

Suitable Grade Level

Grades 10-12