Problem 37 - Algebra and Trigonometry

Question

These problems involve distinguishable permutations. Arrangements In how many ways can two blue marbles and four red marbles be arranged in a row?

Step-by-Step Solution

Answer

The marbles can be arranged in 15 different ways.

Step 1: Understand the Problem

We are asked to find the number of different ways to arrange a total of six marbles, where 2 are blue and 4 are red. This is a permutations problem with distinguishable items.

Step 2: Apply the Formula for Permutations with Distinguishable Objects

The formula to find the number of ways to arrange distinguishable objects is given by \(\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}\), where \(n\) is the total number of objects, and \(n_1, n_2, \ldots, n_k\) are the counts of each type of distinguishable object.

Step 3: Substitute the Values into the Formula

Here, the total number of marbles \(n = 6\) (2 blue and 4 red marbles). So, \(n_1 = 2\) and \(n_2 = 4\). Substitute these values into the formula:\[\text{Ways} = \frac{6!}{2! \times 4!}\]

Step 4: Calculate 6! (Factorial of 6)

Calculate 6 factorial: \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Step 5: Calculate 2! (Factorial of 2) and 4! (Factorial of 4)

Calculate 2 factorial: \(2! = 2 \times 1 = 2\).Calculate 4 factorial: \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 6: Substitute Factorials into the Formula and Solve

Substitute the factorials calculated into the formula:\[\text{Ways} = \frac{720}{2 \times 24} = \frac{720}{48} = 15\]So, there are 15 ways to arrange the two blue marbles and four red marbles in a row.

Key Concepts

Factorial

The term "factorial" refers to the product of a series of descending natural numbers, starting from a specified number down to 1. Factorials are invaluable in permutations and combinations for determining the number of ways in which a set of items can be arranged or selected. The factorial of a number denoted as \( n! \), is calculated by multiplying every integer from \( n \) down to 1. For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)

To contextualize in the given problem, calculating factorials allows us to determine arrangements, such as finding how many different sequences six marbles can form. Remember, the factorial gets large rapidly as \( n \) increases, demonstrating its power in counting principles.

Distinguishable Permutations

Distinguishable permutations deal with arranging items where some items are indistinct from each other. For example, when items of the same type cannot be distinguished, it's important to count only truly unique arrangements. This is valuable in problems involving similar objects, such as marbles of the same color or letters in a word with repeated characters. The formula used to calculate permutations for distinguishable items, \( \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \), involves dividing the total permutations by the factorials of the counts of identical items. The structure helps factor out the identical arrangements and focus on arrangements that stand out as unique. For instance, if there are 6 marbles in total, with 2 blue and 4 red, this formula allows us to find the unique sequences.In the exercise provided, we are interested in finding different sequences with two blue marbles and four red marbles, leading to a total of 15 arrangements by applying this method.

Combinatorial Methods

Combinatorial methods encompass various mathematical techniques used for counting, arranging, and combining objects. These methods are central to topics like permutations, combinations, and probability. They facilitate understanding how to approach complex arrangements and combinations in a simplified, mathematical context. Using these methods allows us to solve problems systematically. For example:

• Permutations focus on arrangements where order matters.
• Combinations are about selections where order does not matter.

In the context of the problem involving marbles, combinatorial methods come into play by utilizing calculations from factorials and permutation formulas to derive meaningful answers. These provide structured approaches to ensure that we account for every possible configuration of items, whether they are similar or unique. Understanding these methods equips students with tools to solve a broad range of mathematical challenges effectively.