Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving combinations. Three financial planners are to be selected from a group of 12 to participate in a special program. In how many ways can this be done? In how many ways can the group that will not participate be selected?

Step-by-Step Solution

Answer

Both participating and non-participating groups can be chosen in 220 ways.

Step 1: Understand the Problem

We have a total of 12 financial planners and we need to choose 3 to participate in a program. This is a combination problem because the order in which we select the planners does not matter.

Step 2: Formulate the Combination Equation

To find the number of ways to choose 3 planners from 12, use 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. In this case, \( n = 12 \) and \( r = 3 \).

Step 3: Calculate Combinations for Participation

Calculate \( \binom{12}{3} \) using the formula for combinations: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Substitute the values to get \( \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} \). Simplify this to get 220.

Step 4: Calculate Combinations for Non-Participation

Since the total group has 12 planners, if 3 are selected to participate, the remaining 9 will not participate. The number of ways to choose the group that will not participate is \( \binom{12}{9} \). Due to the combinations properties, \( \binom{12}{9} = \binom{12}{3} \) because choosing 3 to participate is the same as choosing 9 not to participate. Thus, the number is also 220.

Key Concepts

Combination Formula

When tackling problems where the order of selection is not important, we use the concept of combinations. The combination formula helps us calculate the number of ways to choose items from a larger set without concerning ourselves with the order in which they are chosen. This is crucial in solving many statistical and probabilistic problems.

The combination formula is given by:

• \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Here:

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

This formula works by considering the total permutations of \(n\) items, (signified by \(n!\)), and then dividing by the permutations of the items selected, \(r!\), and the permutations of the remaining items, \((n-r)!\). This ensures that duplicates where the same items are chosen in different orders are not counted multiple times. Understanding this formula is key to mastering combinations.

Participation Selection

The process of participation selection deals with choosing a subgroup from a larger group for a particular purpose or event. In this case, selecting planners for a special program. Using the combination formula, you can find out how many different groups can be formed when choosing 3 planners from a total of 12.

In this exercise, we calculate \(\binom{12}{3}\). This expression asks us to find out the number of ways to select 3 planners out of 12. Following through the combination formula, it simplifies to \(\frac{12 \times 11 \times 10}{3 \times 2 \times 1}\). This results in 220 different possible groups of planners that can be selected.

This step highlights the practicality of combinations: no matter how large the initial group, you can use them to determine precisely how many smaller groups can be formed without worrying about the order of selection.

Non-Participation Selection

Non-participation selection is the process used to determine the selection of a subgroup that will not participate in a given activity. This concept is particularly important when the number of non-participants is relevant to your calculations. In some cases, this could be more direct than calculating the participating group.

The property of combinations tells us that choosing a group of 3 planners from 12 to participate is the same as choosing which 9 planners will not participate. The formulaic representation of this is \(\binom{12}{9} = \binom{12}{3}\). Due to symmetry in the combination formula, these values are inherently equal since both yield groups of the remaining number not chosen.

Thus, the number of ways the non-participating group can be selected is also 220. This shows how understanding the properties of combinations allows you to solve selection problems from either perspective.

Factorial Calculation

Calculating factorials is a fundamental part of working with the combination formula. A factorial, represented by the exclamation mark, \(n!\), is the product of all positive integers up to \(n\). If you're new to the concept of factorials, think of it as a repeated multiplication of descending numbers from \(n\) down to 1.

For example, \(3! = 3 \times 2 \times 1 = 6\). Factorial calculations are integral to the combination process because they eliminate order and ensure that each group selection combination doesn't double-count by distinguishing order.

In our example, combining factorials, we have \(12!\) for the selection down to \(3!\) and \(9!\) for the selected group not participating. Understanding how to calculate these correctly is crucial to derive the correct number of combinations and is a valuable tool in many areas of mathematics.