Problem 28 - Calculus for Biology and Medicine

Question

A bookstore has 300 science fiction books. Molly wants to buy 5 of the 300 science fiction books. How many selections are there?

Step-by-Step Solution

Answer

There are 20,002,900 ways to select 5 books from 300 science fiction books.

Step 1: Understanding the Problem

Molly wants to select 5 science fiction books from a total of 300 books available at the bookstore. This problem involves a combination selection where the order of selection doesn't matter.

Step 2: Identifying the Formula

When choosing a certain number of items (in this case, 5 books) from a larger group (300 books) without regard to the order of selection, we use the combination formula. The formula for combinations is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Substituting the Values

In this problem, \( n = 300 \) and \( r = 5 \). We substitute these values into the combination formula:\[ C(300, 5) = \frac{300!}{5!(300 - 5)!} \]

Step 4: Calculating the Factorials

First, we simplify the expression by calculating the factorials:\( 300! \) is very large, but can be simplified as \[ 300 \times 299 \times 298 \times 297 \times 296 \] because everything else would cancel out with the factorial in the denominator. Then calculate \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 5: Simplifying the Calculation

Now substitute these into the simplified expression:\[ C(300, 5) = \frac{300 \times 299 \times 298 \times 297 \times 296}{5 \times 4 \times 3 \times 2 \times 1} \] which simplifies to a numerical answer.

Step 6: Final Calculation

Calculating the fraction gives:\[ C(300, 5) = \frac{300 \times 299 \times 298 \times 297 \times 296}{120} \] Performing this calculation results in a large number of possible selections.

Key Concepts

Combination Formula

The combination formula is a key concept in combinatorics used to determine the number of ways to choose a subset of items from a larger set, where the order of selection doesn't matter. It is especially handy when you want to know how many different groups of "r" items can be formed from a larger group of "n" items. The formula is given by:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to be chosen.
• The "!" symbol represents the factorial of a number, which is the product of all positive integers up to that number.

The combination formula is used when the order of selection does not matter, as opposed to permutations where the order does matter. In our original exercise, Molly is choosing 5 books from 300, so it's a perfect scenario for using the combination formula because we are interested in different sets of books, not the order they are chosen in.

Factorials

Factorials play a foundational role in combinatorics, particularly with the combination formula. The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It’s a way of calculating products for sequences of decreasing natural numbers.
For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 0! = 1 \) (by definition)

Factorials grow very quickly, which can lead to some large numbers, e.g., \( 300! \) is incredibly large. In many applications like our example problem, it's often more practical to use simplifications based on cancellation in the formula rather than calculating \( n! \) directly. For permutations and combinations, we only compute the necessary portion of the factorial.

Permutations

Permutations relate to arranging a subset of a larger set of items where the order does matter. It's different from combinations, where the order does not matter. In permutations, each possible arrangement of the items counts as a unique permutation.
The formula for permutations is:

• \[ P(n, r) = \frac{n!}{(n-r)!} \]

Here:

• \( n \) is the total number of items.
• \( r \) is the number of items to arrange.

Understanding permutations helps distinguish exercises where order is a crucial factor. For instance, if Molly needed to choose and arrange the books in a specific order on her shelf, permutations rather than combinations would be the appropriate tool. Unlike combinations, permutations give more importance to the specific sequence of selection or arrangement.