Problem 39 - Calculus for Biology and Medicine

Question

A collection contains seeds for four different annual and three different perennial plants. You plan a garden bed with three different plants, and you want to include at least one perennial. How many different selections can you make?

Step-by-Step Solution

Answer

There are 31 different selections possible.

Step 1: Identify Total Plants and Requirement

We have a total of seven plants: four annuals and three perennials. We want to choose three plants, ensuring at least one perennial is included.

Step 2: Calculate Total Selections Without Restriction

Calculate the total number of ways to select any three plants from the seven available, without considering the perennial restriction. This is done using the combination formula: \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.\[\binom{7}{3} = \frac{7!}{3!(7-3)!} = 35\]

Step 3: Subtract Selections With No Perennials

To ensure at least one perennial, subtract the number of ways to choose only annuals from the total combinations. Choosing three plants from only the four annuals is:\[\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4\]

Step 4: Calculate Final Result

Subtract the combinations using only annuals from the total combinations to satisfy the condition of having at least one perennial.\[35 - 4 = 31\]Thus, there are 31 different selections possible.

Key Concepts

Permutation and Combination

When tackling problems on counting methods, it's important to understand the distinction between permutations and combinations.

• **Permutations** refer to different arrangements of a set of items. Order matters with permutations. For example, arranging 3 books on a shelf is a permutation problem.
• **Combinations**, on the other hand, are selections of items where order does not matter. The garden planting problem is a combination problem because it only matters which plants are selected, not the order they are arranged.

In the garden problem, we're interested in how many ways we can select plants, which makes this a combination problem. The key is selecting from a group without regard for order, which is where combinations shine. The formula to calculate combinations is given by: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose.

In this case, figuring out how many combinations of plants can be chosen involves computing these values properly and understanding the conditions given, such as the requirement to include at least one perennial.

Binomial Coefficient

The binomial coefficient is a crucial concept when dealing with combination problems in combinatorics.

It is denoted using the symbol \(\binom{n}{r}\), which is a representation of combinations. It essentially provides a way to count the number of possible combinations of \(r\) objects from \(n\) total objects without caring for the order. The formula used is:

\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]The factorial operation \( ! \) is used extensively here. For instance, \(7!\) ("7 factorial") is calculated as \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

In the garden example, calculating \(\binom{7}{3}\) helped determine how many unrestricted ways you could select plants. This is then adjusted to fit the requirement of including a perennial, showing the flexibility and importance of the binomial coefficient in solving such problems.

Mathematical Problem Solving

Mathematical problem-solving often involves breaking down the problem into smaller, manageable parts.

By identifying what you know and what you need to find out, you can systematically apply mathematical tools and concepts to reach a solution.

In our garden problem:

• **Identify Known Elements:** We have 4 annual plants and 3 perennials, a total of 7 plants.
• **Understand the Restrictions:** At least one perennial must be part of your selection of three plants.
• **Apply Combination Formulas:** Use the binomial coefficient formula \(\binom{n}{r}\) to find total combinations and refine with restrictions.
• **Subtract Non-Desirable Outcomes:** By calculating all combinations and removing the cases that do not meet the criteria, that is, selections with zero perennials, we reach the desired solution.

Mathematical problem-solving requires logical reasoning, systematic calculation, and often, verifying your results to ensure accuracy. Through practice and application, you'll develop an intuition for approaching and solving these kinds of combinatorial problems effectively.