Question

The number of ways 5 English books and 4 Hindi books can be placed on a shelf so that the books on the same language always remain together is (a) \(5 ! 4 ! 3 !\) (b) \(5 ! 4 ! 2 !\) (c) \(4 ! 3 ! 2 !\) (d) \(5 ! 3 ! 2 !\)

Short answer

(b) \(5! 4! 2!\)

Step-by-step solution

Arrange English books

Firstly, arrange the 5 English books among themselves. The number of ways to arrange n items is given by \(n!\). So the 5 English books can be arranged in \(5!\) ways.

Arrange Hindi books

Then, arrange the 4 Hindi books among themselves. Similarly, the 4 Hindi books can be arranged in \(4!\) ways.

Arrange the batches

Now, consider the batch of English books and the batch of Hindi books as two separate objects. These two objects can be arranged among themselves in \(2!\) ways.

Get the total arrangements

Finally, to get the total number of arrangements, multiply the number of arrangements of the English books, the Hindi books and the two batches among themselves: \(5! * 4! * 2!\)

Key concepts

Factorials

Factorials are a way to calculate the number of ways to arrange a set of objects. Denoted by the symbol "!", a factorial of a number is the product of all positive integers up to that number. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are used extensively in permutations and combinations to determine the number of possible ways to arrange or combine items. They are especially useful in scenarios involving the arrangement of objects, like books on a shelf, seats in a row, or players in a lineup.

Understanding how to calculate factorials is foundational as it serves as a building block for more complex concepts in mathematical problem-solving.

Arrangement of Objects

When we talk about the arrangement of objects, we often refer to permutations. Permutations deal with the different ways to arrange a set of items where the order of arrangement is significant. For example, in arranging books, the order in which they appear matters greatly.

In the given problem, we start by arranging the English books. Given there are 5 of them, they can be arranged in \(5!\) different ways. Next, for the 4 Hindi books, there are \(4!\) possible arrangements.

We now consider each group as a single object or 'block.' The English block and the Hindi block can be further arranged in \(2!\) ways, depending on which block comes first. Thus, the arrangement problem is dissected into manageable sections by considering factorials of each block and then merging the arrangements.

Such problems showcase how permutations and combinations are used to efficiently solve complex arrangement tasks.

Mathematical Problem Solving

Mathematical problem-solving involving permutations and combinations, as demonstrated in the exercise, requires both a methodical and creative approach. The problem of arranging books on a shelf, while seeming straightforward, involves layers of decision-making.

1. **Break Down the Problem:** Divide the problem into smaller steps. Here, we separate English and Hindi books, treating each group initially by itself.

2. **Use Factorials Wisely:** Factorials help calculate arrangements. Understanding how to apply them to both individual items and groups is crucial.

3. **Combine Solutions:** After individual calculations, the final step involves combining all elements to find the total possibilities, such as \(5! \times 4! \times 2!\).

4. **Double-check Logic:** We verify steps by looking at the total arrangement and ensure each possibility is accounted for accurately.

This problem-solving approach, where complexity is simplified using logical steps and mathematical tools, builds strong analytical skills necessary for tackling a variety of mathematical challenges.