Problem 50 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Choosing Clothing If a woman has eight skirts, in how many ways can she choose five of these to take on a weekend trip?

Step-by-Step Solution

Answer

There are 56 ways.

Step 1: Understand the Problem

We need to determine how many ways a woman can choose 5 skirts from a collection of 8. This is a combinations problem because the order of selection does not matter.

Step 2: Recall 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: Substitute Values into the Formula

Here, \( n = 8 \) (total skirts) and \( r = 5 \) (skirts to choose). Substitute these values into the combination formula:\[ C(8, 5) = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \times 3!} \]

Step 4: Calculate the Factorials

Evaluate the factorials:\( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)\( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)\( 3! = 3 \times 2 \times 1 \)

Step 5: Simplify the Expression

Calculate the expression:\[ C(8, 5) = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} \]The \( 5! \) terms cancel out, leaving:\[ C(8, 5) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \]

Step 6: Perform the Simplification and Calculation

First, calculate the numerator:\( 8 \times 7 \times 6 = 336 \)Next, calculate the denominator:\( 3 \times 2 \times 1 = 6 \)Now divide:\[ C(8, 5) = \frac{336}{6} = 56 \]

Step 7: Final Solution

The number of ways the woman can choose 5 skirts from 8 is 56.

Key Concepts

Combination Formula

In mathematics, when you need to find out how many ways you can choose items from a larger set without considering the order, the combination formula is your best friend. This formula allows you to calculate combinations, which is why it doesn't account for the order of selection—thus differing from permutations.
Let's take a closer look at the formula used to calculate combinations:

• Formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \)
• Explanation: In the formula, \( n \) represents the total number of items you have—in this exercise, that's 8 skirts.
• \( r \) is the number of items you want to choose—in our case, 5 skirts for the weekend trip.

The combination formula effectively counts the number of groups (combinations) you can create from the total set without paying attention to the order in which they appear. It’s a vital tool in combinatorics, especially when you want to plan without regard to sequence.

Factorial Calculation

The factorial calculation is represented by an exclamation mark (!) and involves multiplying a series of descending natural numbers. This method is foundational to understanding combinations because it forms part of the combination formula. For example, to calculate how many ways you can organize a specific group of items, you calculate the factorial of a number.

• Definition: The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).
• Example: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

In our problem, you use factorials to compute the number of ways to choose 5 skirts out of 8. You need to calculate \( 8! \), \( 5! \), and \( 3! \), where \( 3! = 3 \times 2 \times 1 \). These calculations simplify parts of the combination formula and make solving the exercise much easier and organized.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and finding patterns within sets of elements. It is essential in solving numerous real-life problems where the arrangement and selection of objects are crucial.

• Key Application: In our exercise, combinatorics helps us determine the number of ways to choose items without considering order. This process is critical in scenarios such as lottery drawings, classifying objects, or planning trips (like selecting the best skirts to bring along).
• Core Problem-Solving Tool: Using combinatorics efficiently enables you to determine how to structure or manage different scenarios where selection matters.

The core idea is that if the sequence or arrangement doesn't matter, combinations give you the right structure to solve the problem efficiently and logically. As shown in our skirt-choosing question, you're only interested in which skirts go on the trip, not in what order they're packed!