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 10 different ones.

Step 1: Understanding the Problem

We need to find the number of ways to choose 3 candy bars from a total of 10 different candy bars. This is a combinations problem, where the order of selection does not matter.

Step 2: Using the Combinations Formula

The formula for 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. In this case, \( n = 10 \) and \( r = 3 \).

Step 3: Substituting Values into the Formula

Substitute \( n = 10 \) and \( r = 3 \) into the combinations formula:\[ C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \, 7!} \]

Step 4: Calculating the Factorials

Calculate the factorials:- \( 10! = 10 \times 9 \times 8 \times 7! \)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( 7! \) in the numerator and denominator cancels out.

Step 5: Performing the Division

Divide the remaining values:\[ C(10, 3) = \frac{10 \times 9 \times 8}{6} \]First calculate the numerator: \( 10 \times 9 \times 8 = 720 \).Then divide by 6: \( \frac{720}{6} = 120 \).

Key Concepts

Permutations and Combinations

When selecting items, understanding whether or not the order of selection matters leads us to use either permutations or combinations.
In our candy bar problem, the order does not matter. As long as we have 3 candy bars, the sequence is irrelevant. That's where combinations are used.
Permutations on the other hand, consider different orders as distinct outcomes. This is typically needed when arrangements or sequences need to be considered.

• **Permutations**: Focuses on arrangements where order matters.
• **Combinations**: Centers on selections where order does not matter, just like our candy selection scenario.

In our bag of candy bars, choosing 3 out of 10 different bars brings us to a classical combinations scenario. Here, we do not care about which candy bar is chosen first or last, just that we have picked 3 total.
This drives us to use the combinations formula where each choice counts once.

Factorials

Factorials are fundamental in combinatorial problems as they simplify complex multiplication across a sequence of consecutive integers.
A factorial, denoted by an exclamation mark, multiplies a series of descending natural numbers.

• **For example,**: The factorial of 4, written as \( 4! \), is calculated as \( 4 \times 3 \times 2 \times 1 = 24 \).

In our candy bar decision, calculating combinations involved factorials in the formula.
The expression \( n! \) is essential as it represents all possible arrangements for any n items. Since we're choosing 3 out of 10 candy bars, the usage of \( 10! \), \( 3! \), and \( 7! \) facilitates cancellation and simplification in this problem.
Here, the factorial \( 7! \) occurs both in the numerator and denominator allowing us to easily reduce the calculation into fewer multiplications.
Factorials in combinations help manage the reduction of repetitive calculation by neatly simplifying the problem.

Combinatorial Counting

Combinatorial counting involves counting the number of ways to arrange or select items under a set of rules.
In the candy bar exercise, our task is to count how many ways we can choose 3 candy bars from a mix of 10, focusing only on selections and not sequence.
The art of combinatorial counting lies in recognizing boundaries set by the problem's context, such as order relevance or constraints.

• **Combinations Formula**: \( C(n, r) = \frac{n!}{r!(n-r)!} \) is used explicitly when the order is irrelevant.
• **Identifying Constraints**: Understanding what you're counting helps define whether combinations or permutations apply.

In this particular case, following the methodical approach of writing, substituting values in, and simplifying leads to \( 120 \) as our answer—meaning there are 120 unique ways to choose the candy bars.
Combinatorial counting offers clearer insight into a problem frequently encountered in probability and statistics, aiding in deeper analytical thinking.