Problem 54 - Algebra and Trigonometry

Question

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 to choose five skirts from the eight.

Step 1: Identify the Problem Type

This problem involves combinations, where order does not matter, and we are choosing a subset of items from a larger set. We need to use the combinations formula.

Step 2: Recall the Combinations Formula

The formula for finding the number of combinations of choosing \( r \) items from \( n \) items is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n! \) is the factorial of \( n \).

Step 3: Identify Values for the Formula

In this problem, \( n = 8 \) (the total number of skirts) and \( r = 5 \) (the number of skirts she wants to choose).

Step 4: Apply the Values to the Formula

Substitute the values into the combinations formula: \( \binom{8}{5} = \frac{8!}{5!(8-5)!} \).

Step 5: Calculate the Factorials

Calculate the factorials: \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \), and \( 3! = 3 \times 2 \times 1 \).

Step 6: Simplify the Expression

Solve for \( \binom{8}{5} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1 \times 3 \times 2 \times 1} \). Cancel out common terms in the numerator and denominator to simplify.

Step 7: Perform Simplified Calculation

Simplify: \( \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 8 \times 7 = 56 \).

Step 8: State the Final Answer

There are 56 different ways for her to choose five skirts from eight.

Key Concepts

Combinations Formula

When faced with situations where you need to select a subset from a larger set without caring about the order, combinations are the way to go. This concept helps in finding the number of different ways to choose items from a larger group. The key tool here is the combinations formula, which is typically represented as \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

• \( n \) represents the total number of items.
• \( r \) stands for the number of items you want to choose.
• Factorials, denoted by \(!\), will be discussed in detail in the next section.

This formula is crucial when the problem mentions terms like "choose", "pick", or "select" without regard to order. In our clothing example, since the woman wants to select skirts irrespective of order, we use the combinations formula.

Factorial Calculation

Factorial calculation is a fundamental concept in combinations. It involves multiplying a series of descending natural numbers. For instance, the factorial of 8, written as \(8!\), would be calculated as \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \].Factorials simplify the computation of combinations by allowing you to easily calculate the needed permutations and then divide to get combinations.Here are some factorial points to remember:

• \( n! \) grows quite fast as \( n \) increases, so always break it down step-by-step.
• The formula breakdown quickly reveals which terms can be cancelled out.
• For small numbers, it's easier to calculate directly by hand.

Recognizing where factorials can be shared or canceled can simplify calculations a lot, reducing the chance of errors.

Problem-Solving Steps

When tackling a combinations problem, a systematic approach helps guarantee success. The structured, step-by-step method employed in our skirt selection problem ensures clarity and precision: 1. **Identify**: Recognize if the problem concerns combinations—you're often prompted by phrases like "choose" or "pick" without any mention of order. 2. **Recall the Formula**: Reaffirm the combinations formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) and understand its components. 3. **Substitute Values**: Insert your given numbers—like \( n = 8 \) and \( r = 5 \)—into the formula. 4. **Calculate Factorials**: Compute the factorials for given numbers. Factor crowns like \( 5! \) break into manageable parts and help in simplifying. 5. **Simplify and Solve**: Reduce the equation by cancelling out common terms, making the calculation easier. Following these steps not only clarifies one's thought process but also reduces possible missteps. Learning to embrace these steps creates a reliable framework for tackling more complex combination challenges.