Problem 53 - College Algebra

Question

Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?

Step-by-Step Solution

Answer

The number of distinct arrangements is \( \frac{20!}{6! \times 6! \times 8!} \).

Step 1: Understand the Problem

Susan has 20 plants in total, made up of 6 tulips, 6 roses, and 8 daisies. We need to find the number of distinct ways to arrange these plants.

Step 2: Calculate Total Arrangements Without Restrictions

First, assume that all plants are distinct. Therefore, the total number of arrangements would be the factorial of the total number of plants: \[ 20! \]

Step 3: Account for Indistinguishable Plants

Since tulips, roses, and daisies of the same type are indistinguishable among themselves, we must divide by the factorials of the count of each type to correct the overcounting. This gives us the formula:\[ \frac{20!}{6! \times 6! \times 8!} \]

Step 4: Calculate Each Factorial

Compute the factorial for each number:- \(20!\) is the factorial of 20- \(6!\) is 720- \(8!\) is 40320.

Step 5: Plug in Values and Solve

Substitute the calculated factorials into the formula: \[ \frac{20!}{6! \times 6! \times 8!} = \frac{20!}{720 \times 720 \times 40320} \] Perform the division to find the number of distinct arrangements.

Key Concepts

Factorial

In the world of combinatorics, the concept of a factorial is a fundamental building block. A factorial is denoted by an exclamation mark, like this: \[ n! \]It represents the product of all positive integers from 1 up to the number \( n \). For example, \( 5! \) means

• 5 x 4 x 3 x 2 x 1, which equals 120.

Calculating factorials is crucial for determining the total number of distinct arrangements or permutations of a set of items. When you're trying to arrange 20 plants in a row, you would, in theory, want to calculate \( 20! \). This number represents all possible ways the plants could be arranged—in other words, the "unrestricted" arrangements. Understanding this foundational concept allows us to delve deeper into more complex combinatorial scenarios, such as considering indistinguishable objects.

Permutations

Permutations refer to the different ways we can arrange a set of items where order matters. If all items were distinct, calculating permutations is as simple as using the factorial of the total number of items, like \[ 20! \] for 20 distinct plants. But in reality, permutations become more interesting and complex when some items are indistinguishable.When dealing with permutations, one key formula emerges: for a set of \( n \) items where some are identical, the total number of unique permutations is given by \[\frac{n!}{k_1! \times k_2! \times \ldots \times k_r!}\]This formula accounts for the indistinguishable items, denoted by \( k_1, k_2, \ldots, k_r \), which are the counts of each type of identical item. In the garden scenario, since we have 6 tulips, 6 roses, and 8 daisies, the formula becomes \[\frac{20!}{6! \times 6! \times 8!}\]Thus, by calculating this, we recognize how permutations adapt when faced with indistinguishable items, providing a more accurate count of unique arrangements.

Indistinguishable objects

In many real-world scenarios, we encounter sets where some objects are indistinguishable from others. This occurrence complicates pure permutation calculations because, without adjustment, we would overcount identical arrangements. In simpler terms, those tulips, roses, and daisies that are identical must be accounted for by dividing by their factorials.Given our example with 20 plants, we must consider the types of plants:

• 6 identical tulips
• 6 identical roses
• 8 identical daisies

The mathematical solution to distinguish these unique permutations within the total count uses the formula:\[\frac{n!}{k_1! \times k_2! \times \ldots \times k_r!}\]Here, \( n \) stands for the total number of plants, and \( k_1 \), \( k_2 \), etc., are the counts of each identical object. This correction is essential to achieve the right number of unique arrangements. For instance, without considering indistinguishable objects in our 20 plants, we could mistakenly believe many more unique sequences are possible. Through this understanding, mathematical accuracy shines, turning vast possibility into counted certainty.