Problem 19 - Calculus for Biology and Medicine

Question

A bag contains 10 different candy bars. You are allowed to choose \(3 .\) How many choices do you have?

Step-by-Step Solution

Answer

There are 120 ways to choose 3 candy bars from the bag.

Step 1: Understanding the Problem

We need to determine the number of ways to choose 3 candy bars from a group of 10 different options. This is a problem of combinations, as the order in which we select the candy bars doesn't matter.

Step 2: Setting Up the Combination Formula

The formula to calculate combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. For this problem, \( n = 10 \) and \( r = 3 \).

Step 3: Applying the Formula

Substitute the values into the formula: \( C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} \).

Step 4: Simplifying the Factorials

Calculate the factorials: \( 10! = 10 \times 9 \times 8 \times 7! \). The \( 7! \) terms cancel out in the fraction, simplifying to \( \frac{10 \times 9 \times 8}{3!} \).

Step 5: Calculating the Result

Compute \( 3! = 3 \times 2 \times 1 = 6 \). Then simplify the expression: \( \frac{10 \times 9 \times 8}{6} \). This converts to \( \frac{720}{6} = 120 \).

Step 6: Conclusion

Therefore, there are 120 different ways to choose 3 candy bars from the bag.

Key Concepts

Factorials

Factorials are a mathematical concept used to represent the product of all positive integers up to a given number. They are an essential tool in many areas of mathematics, including permutations and combinations. The factorial of a number "n" is denoted as \( n! \). It is calculated by multiplying all whole numbers from the given number down to 1. This means for any positive integer \( n \):
\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \]
For example, the factorial of 4, written as \( 4! \), is 4 times 3 times 2 times 1; that is, \( 4! = 24 \).

Factorials grow very rapidly as the number increases. For instance, \( 5! = 120 \) and \( 6! = 720 \). This rapid growth is why we often encounter already large numbers even for relatively small initial values.

In problems involving combinations and permutations, factorials help in counting arrangements and groupings. They allow us to quickly find how many different ways a set number of items can be arranged or chosen from a group.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and combining items within certain constraints. It helps us solve questions like the one about choosing candy bars from a bag. In combinatorics, two of the most common concepts are permutations and combinations. These concepts help determine how items from a set can be arranged or selected.

When dealing with combinations, which is the case in our candy bar problem, we are concerned with the number of ways to choose a subgroup from a larger set, where the order does not matter.
In combinatorics, the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) is pivotal:

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

This formula helps compute scenarios where the order of selection is irrelevant, as in picking candy bars: \( C(10, 3) = \frac{10!}{3!7!} \).

The field of combinatorics goes beyond just this and examines deeper into graphs, sequences, and arrangements, making it a versatile and crucial area in mathematics.

Permutations

Permutations refer to the number of different ways to arrange items in a specific order. Unlike combinations, in permutations, the order in which items are arranged does matter. For example, if you are lining up 3 different candy bars from a selection of 10, permutations would count each specific arrangement separately.

The formula for permutations is \( P(n, r) = \frac{n!}{(n-r)!} \):

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

For instance, if we wanted to arrange 3 candy bars out of 10, we'd calculate it using \( P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} \). As you can see, each arrangement of the candy bars would be counted separately.

Permutations are used in many applications such as scheduling, cryptography, and optimizing routes. They show us all the possible ways to position items where sequence matters, offering a detailed look into combination possibilities.