Problem 39 - Algebra and Trigonometry

Question

Eight mathematics books and three chemistry books are to be placed on a shelf. In how many ways can this be done if the mathematics books are next to each other and the chemistry books are next to each other?

Step-by-Step Solution

Answer

Total ways: 2! x 8! x 3! = 1,933,440 ways.

Step 1: Group Mathematics Books

Since the mathematics books need to be next to each other, we can treat them as a single unit or 'block.' Thus, we have one mathematics block, and its internal arrangements can be calculated separately.

Step 2: Group Chemistry Books

Similarly, the chemistry books need to be next to each other, so we also treat them as a single unit or 'block.' Calculate its internal arrangements separately as well.

Step 3: Count Arrangements of Blocks

Now, we have two blocks (mathematics and chemistry), plus the other blocks formed by treating each group as a single unit. So, we arrange these two "blocks" on the shelf. The arrangement of these blocks is treated as arranging two distinguishable items.

Step 4: Arrange Within Blocks

Within the mathematics block, the 8 books can be arranged in any order. Therefore, we calculate the number of ways to arrange 8 books as \(8!\). Similarly, within the chemistry block, the 3 books can be arranged in any order, giving us \(3!\) arrangements.

Step 5: Calculate Total Arrangements

The total arrangements are calculated by multiplying the arrangements of the blocks with the permutations within each block. Therefore, the formula is: \[2! \times 8! \times 3!\] Compute this to get the total number of ways to arrange the books.

Key Concepts

Factorial

The concept of a factorial is crucial when calculating permutations of distinct objects. A factorial, represented by an exclamation mark at the end of a number (like 5!), simply means you multiply that number by every whole number below it, down to 1. For example,

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 3! = 3 × 2 × 1 = 6
• 1! = 1

When arranging items, each unique arrangement is one of the permutations. In our problem, we use factorials to determine how many ways we can arrange the mathematics books and chemistry books once grouped into blocks. This is why we calculate 8! for the mathematics books and 3! for the chemistry books. Factoring in 2! helps us see how to arrange the two main blocks on the shelf.

Block Arrangement

In solving permutation problems, block arrangement is a useful approach when dealing with grouped items that need to stay together. This method treats a group of items, such as a stack of books, as a single unit or block when calculating placements.

For instance, the eight mathematics books are treated as one block, and the three chemistry books make another block. The role of block arrangement simplifies the problem since it reduces the number of items from 11 individual books to just 2 larger blocks.

• This technique can easily transform a more complex arrangement problem into a simpler one with fewer variables.

Thus, by initially arranging the blocks themselves and then arranging the items within each block afterward, the overall calculation becomes more manageable and clear.

Mathematics Books Arrangement

Once the mathematics books are grouped into a single block, we can start thinking about how to arrange the books within this block. With eight mathematics books, each one can be rearranged among themselves in different ways.
To find out how many unique ways you can rearrange them, you use the concept of a factorial. Specifically here, you calculate it using 8!, which means:

• 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

This high number represents the permutations of the eight mathematics books if they are all stacked next to each other. Thus, regardless of how you place the blocks on the shelf, within their own group, the books can be flipped, swapped, and rearranged in many different combinations, which is a classic example of the power of permutations and arrangements.

Chemistry Books Arrangement

Similarly to the mathematics books, the chemistry books are also grouped into a single block. For these three books, the permutation within the block is calculated using the factorial as well.
The arrangement within the chemistry block is determined by 3!, or:

• 3 × 2 × 1 = 6

This solution tells us there are 6 different ways to rearrange the three chemistry books once grouped.
Understanding this block's arrangement within itself is critical because, even though they need to stick together on the shelf, the internal order can shift, allowing for these six distinct arrangements total.