Problem 34 - Algebra and Trigonometry

Question

33–40 These problems involve distinguishable permutations. Arrangements In how many different ways can five red balls, two white balls, and seven blue balls be arranged in a row?

Step-by-Step Solution

Answer

There are 14,249,100 ways to arrange the balls in a row.

Step 1: Understanding the Problem

To solve this problem, we need to determine how many different ways the balls can be arranged in a row given that there are objects of the same type (red, white, and blue balls). This requires calculating permutations of distinguishable items.

Step 2: Applying Permutation Formula for Multisets

The formula for permutations of a multiset is given by \( \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} \), where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) are the counts of each type of item. Here, we have 5 red, 2 white, and 7 blue balls.

Step 3: Calculating the Total Number of Balls

First, calculate the total number of balls: \( 5 + 2 + 7 = 14 \). So, there are 14 balls in total.

Step 4: Calculate Factorials of Counted Items

Compute the factorial of the total items and each subgroup. Total factorial is \( 14! \), red balls have \( 5! \), white balls have \( 2! \), and blue balls have \( 7! \).

Step 5: Substitute into the Permutation Formula

Substitute these factorials into the formula: \( \frac{14!}{5! \cdot 2! \cdot 7!} \).

Step 6: Compute the Permutation

Calculate the expression: \[ \frac{14!}{5! \cdot 2! \cdot 7!} = \frac{87,178,291,200}{120 \cdot 2 \cdot 5,040} \]This equals 14,249,100.

Key Concepts

Multisets

Imagine you have a basket filled with different colored balls. Instead of each ball being unique, some colors repeat. This collection of balls is called a multiset. Unlike regular sets where each element is unique, multisets allow for repetition. You'll encounter multisets often in combinatorial problems where you're asked to count arrangements or group objects. For example in our problem, the multiset consists of five red, two white, and seven blue balls. Multisets challenge us to think differently about counting, since the repeated items require special attention in permutations and combinations. Using the right formula, we can account for these repetitions effectively.

Permutation Formula

Permutations help us figure out how to arrange items where order matters. But when it comes to multisets, we need a special permutation formula. The general formula is:

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

Here, \( n \) is the total number of items and \( n_1, n_2, \ldots, n_k \) are the frequencies of each distinct item in the multiset. To solve our specific problem, we plug in our totals: 14 balls in total, with 5 red, 2 white, and 7 blue. This formula takes care of the repeated items, making sure we don’t count the same arrangement more than once. Understanding this formula is key when working with problems involving repeated items.

Factorials

Factorials are essential in permutations and combinations. The factorial of a number, denoted \( n! \), is the product of all positive integers up to \( n \). It tells us how many ways \( n \) distinct items can be arranged. For example, 3! = 3 \times 2 \times 1 = 6.
In our problem, we compute several factorials:

• \( 14! \) for all balls
• \( 5! \) for red balls
• \( 2! \) for white balls
• \( 7! \) for blue balls

These calculations help us apply the permutation formula effectively, canceling out duplicated arrangements from repeated items. Remember, factorials grow very fast, so calculating them becomes challenging for large numbers, but they are vital for accurate combinatorial computations.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting combinations and permutations. It helps us understand how items can be selected and arranged. When tackling problems like arranging colored balls, combinatorics offers the tools and formulas needed to find solutions.
For our exercise, combinatorics guides us through using permutations to count how many distinguishable ways we can arrange a multiset.

• Identify the total set and subsets
• Apply permutation formulas for repeated items
• Use factorials to compute possible arrangements

By leveraging combinatorics, we transition from a seemingly complex problem to a set of methodical steps revealing the solution. This area of mathematics shows us that even in large sets or where items repeat, we can still find order and compute exact numbers of arrangements.