Question

There are 3 copies each of 4 different books. The number of ways they can be arranged in a shelf is (a) 369600 (b) 400400 (c) 420600 (d) 440720

Short answer

The total number of ways to arrange 12 books on a shelf, with 4 different books having 3 identical copies each, is 369600. To find this, we used the permutation formula with repetition: \( \frac{n!}{n_1! \times n_2! \times ... \times n_k!} \), where n = 12 and \(n_1\) = \(n_2\) = \(n_3\) = \(n_4\) = 3. After calculating the factorials and plugging in the values, we found the answer to be (a) 369600.

Step-by-step solution

Understand the problem

There are 12 books to be arranged in a shelf, with 4 different books having 3 identical copies each. Our task is to find the total number of ways to arrange these 12 books on the shelf.

Apply the permutation formula with repetition

To find the total number of arrangements, we will use the permutation formula with repetition: \( \frac{n!}{n_1! \times n_2! \times ... \times n_k!} \) Here, n is the total number of objects (books), and \(n_i\) represents the number of repetitions of each object. In this case, n = 12 and \(n_1\) = \(n_2\) = \(n_3\) = \(n_4\) = 3.

Calculate the Factorials

Next, we need to calculate the factorials of the numbers involved in the formula: \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600\) \(3! = 3 \times 2 \times 1 = 6\)

Calculate the number of arrangements

Plug in the values from Step 3 into the permutation formula with repetition: \( \frac{479001600}{6 \cdot 6 \cdot 6 \cdot 6} = \frac{479001600}{1296} = 369600 \)

Choose the correct option

According to our calculation, there are 369600 ways to arrange the books on the shelf. Thus, the answer is: (a) 369600

Key concepts

Factorial Calculations

Factorial calculations are a fundamental part of combinatorics and permutations. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

When dealing with permutations and arrangements, factorials help to account for all possible orderings of a set of items. The calculation of large factorials, like \(12!\) in the given exercise, is typically done step by step, multiplying each successive integer down to 1. The factorial of a number \(n\) represents the total number of ways to arrange \(n\) distinct items in sequence.

Understanding the concept of factorial is crucial because it lays the groundwork for more complex probability and permutation problems, where a deep understanding of how to calculate and interpret factorial values is needed to determine the number of possible arrangements or outcomes in a given scenario.

Arrangement of Identical Items

Arranging identical items brings a unique aspect to combinatorial problems. When objects are identical, the order of arrangement among these identical items doesn't matter, simplifying the total count of arrangements.

Consider the exercise where we have multiple copies of identical books. If we treated each of these identical copies as distinct, the number of arrangements would be the factorial of the total number. But since we have identical items, we must divide by the factorial of the number of identical items to remove the arrangements that are essentially the same.

In practice, if we have \(t\) total items and a group of \(n\) identical items, we find the total number of unique arrangements by dividing \(t!\) by \(n!\). This is why, in the exercise, for each set of 3 identical books, we divide by \(3!\) to get the correct number of distinct arrangements. This modification of the basic permutation formula is essential to account for the repetition of items, ensuring each unique arrangement is only counted once.

Counting Principles in Mathematics

Counting principles in mathematics, including the Basic Counting Principle and permutations, provide strategies for determining the total number of possibilities in various scenarios. The Basic Counting Principle states that if one event can occur in \(m\) ways and a second independent event can occur in \(n\) ways, then the two events can occur in \(m \times n\) ways.

Permutations extend this idea to account for the order of arrangement, which is crucial when the sequence is important. The permutation formula with repetition, as seen in the exercise, modifies the basic permutation to accommodate for identical items, thus adhering to the general counting principle that each possible arrangement or outcome must be counted once and only once.

In the context of the exercise, by using the permutation formula with repetition and incorporating the factorial calculations, we've utilized these counting principles to find that the 12 books can be arranged in 369600 ways on a shelf, considering the constraints of identical copies. This application highlights how counting principles are interwoven with other mathematical concepts to solve complex combinational problems.