Problem 74 - College Algebra with Modeling and Visualization

Question

A professor has 3 copies of an algebra book and 4 copies of a calculus text. How many distinguishable ways can the books be placed on a shelf?

Step-by-Step Solution

Answer

35 ways.

Step 1: Total Books Calculation

Count the total number of books. We have 3 algebra books and 4 calculus books, hence: \[ 3 + 4 = 7 \] So, there are a total of 7 books.

Step 2: Formula for Distinguishable Arrangements

To find the number of distinguishable arrangements when there are identical items, use the formula: \[ \frac{n!}{n_1! \times n_2!} \]where \(n\) is the total number of items, and \(n_1\), \(n_2\),... are the number of identical items.

Step 3: Apply the Formula

Apply the formula from Step 2 where \( n = 7 \) (total books), \( n_1 = 3 \) (algebra books), and \( n_2 = 4 \) (calculus books): \[ \frac{7!}{3! \times 4!} \]

Step 4: Calculate Factorials

First, calculate the factorials:- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)- \(3! = 3 \times 2 \times 1 = 6\)- \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Step 5: Final Calculation

Substitute the factorial values into the formula:\[ \frac{5040}{6 \times 24} = \frac{5040}{144} = 35 \]

Step 6: Conclusion

Therefore, there are 35 distinguishable ways to arrange the 3 algebra books and 4 calculus books on the shelf.

Key Concepts

Factorials

Factorials are a fundamental concept when dealing with permutations, combinations, and many other areas in mathematics. A factorial, represented by an exclamation point "!", is the product of all positive integers up to a given number. For example, the factorial of 7, written as \( 7! \), is calculated as \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).

Factorials help in determining the total number of ways to arrange a set of items. This is because each number in the factorial represents a position to be filled by an item. The concept of factorials makes it easier to solve problems involving sequences and arrangements by reducing complex multiplications into a single factorial value.

When you encounter problems involving permutations or combinations, calculating factorials is often one of the first steps needed to find a solution. It's crucial to master this calculation as it forms the base for solving more intricate mathematical problems.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and finding patterns among sets of elements. It's a key tool in solving problems where the arrangement of objects is important, like in the textbook problem we discussed.

One important formula in combinatorics for finding the number of distinguishable arrangements of items is the permutation formula tailored for identical items. The formula is:

• \( \frac{n!}{n_1! \times n_2! \times \ldots} \)

Here, \( n \) is the total number of items, and \( n_1, n_2, \ldots \) are the numbers of identical items within the set. By dividing the factorial of the total number of items by the factorials of the identical items, we account for the indistinguishable swaps among identical items. This helps avoid over-counting arrangements that are actually the same.

Using combinatorics effectively enables us to count possibilities and manage scenarios in both everyday examples and complex mathematical theories.

Identical Items

When dealing with permutations, the presence of identical items requires special attention. Placing identical items into a sequence adds complexity because identical items can be swapped without creating a new arrangement.

In the problem of arranging books, the algebra books are identical to each other, as are the calculus books. This means that simply calculating the factorial of the total number of books would overestimate the number of unique arrangements. To manage this, we use the formula for distinguishable permutations:

• \( \frac{n!}{n_1! \times n_2!} \)

This formula helps eliminate over-counting by dividing the total arrangements by the factorial of each set of identical items, thus giving the accurate count of unique arrangements.

Understanding the concept of identical items is crucial in solving problems about permutations, as it ensures precise calculation of distinct sequences.