Problem 26 - Calculus for Biology and Medicine

Question

Suppose that you want to plant a flower bed with four different plants. You can choose from among eight plants. How may different choices do you have?

Step-by-Step Solution

Answer

There are 70 different ways to choose 4 plants from 8.

Step 1: Understanding the Problem

We need to select 4 different plants from a total of 8 available plants. This is a combination problem as the order of selection does not matter.

Step 2: Apply the Combination Formula

We use the combination formula to find the number of ways to choose 4 plants from 8. The combination formula is given by: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where \(n\) is the total number of plants to choose from, and \(r\) is the number of plants to choose.

Step 3: Substitute Values

Substitute \(n = 8\) and \(r = 4\) into the formula:\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

Step 4: Simplify the Factorials

Calculate the factorials:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Step 5: Calculate the Combination

Now, substitute the values from Step 4 back into the combination formula:\[ \binom{8}{4} = \frac{40320}{24 \times 24} = \frac{40320}{576} = 70 \]

Step 6: Conclusion

There are 70 different ways to choose 4 plants from a selection of 8.

Key Concepts

Factorials

Factorials are an essential mathematical concept that often appears in permutations and combinations. A factorial, represented by the symbol "!", is a product of all positive integers up to a specified number. For a number `n`, the factorial of `n` is denoted as `n!`. This means you multiply all the whole numbers from `n` down to 1.
Here’s how factorials work:

• For any positive integer `n`, the factorial is calculated as: \[ n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \]For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• It’s important to note that the factorial of 0 is defined as 1, i.e., \(0! = 1\).

Understanding how to calculate factorials is crucial for solving problems in combinatorics, particularly when using the combination formula.

Combination Formula

In mathematics, the combination formula is used when we need to find how many ways we can choose a subset of items from a larger set, where the order of selection does not matter. This is known as a combination.
The formula for combinations is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here’s how to break it down:

• `n` stands for the total number of items to choose from.
• `r` represents the number of items we want to select.

For example, if you want to select 4 plants from a collection of 8 different kinds, `n` would be 8, `r` would be 4, and you would use the combination formula to find the total number of possible selections:\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]Calculating this results in 70 possible combinations. This formula is key in solving selection problems where the arrangement of selected items doesn’t matter.

Selection Problem

A selection problem in mathematics involves choosing a specific number of items from a larger set without regard to the order in which they are picked. This is a classic problem in combinatorics, a branch of mathematics dealing with counting and arrangement.
When faced with a selection problem:

• First, determine if the order of selection matters. If not, you will likely use combinations rather than permutations.
• Second, decide on the total number of items (`n`) you have and how many items (`r`) you need to pick.
• Finally, apply the combination formula, which suits these selection problems where order does not matter.

For instance, in our example problem, you are choosing 4 plants out of 8 where order does not matter, making it a selection problem suitable for combination calculations. Such problems can often be seen in real-life scenarios like seating arrangements, choosing teams, or selecting books from a library shelf.