Problem 29 - Algebra 2

Question

CHORES The five children of the Blanchard family get weekly chores assigned to them at random. Their parents put pieces of paper with the names of the five children in a hat and draw them out. The order of the names pulled determines the order in which the children will be responsible for sorting laundry for the next five weeks. What is the probability that Jim will be responsible for the first week and Emily will be responsible for the fifth week?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{20}\).

Step 1: Determine Total Number of Outcomes

First, calculate the total number of ways to assign the chores in order for the five children. This can be done by calculating the permutations of the five children, which is denoted by 5 factorial (5!). The formula is \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Thus, there are 120 different ways to assign chores in a sequence to the five children.

Step 2: Determine Favorable Outcomes for Specific Assignment

Now, let's calculate the number of favorable outcomes where Jim is chosen for the first week and Emily is chosen for the fifth week. If Jim is assigned the first week, Emily the fifth week, we have three remaining weeks and three children to fill those weeks. This can be arranged in \(3!\) different ways. \(3! = 3 \times 2 \times 1 = 6\). So, there are 6 favorable outcomes.

Step 3: Calculate the Probability

The probability is calculated by taking the number of favorable outcomes divided by the total number of outcomes. Therefore, the probability \(P\) that Jim is responsible for the first week and Emily is responsible for the fifth week is \(P = \frac{6}{120} = \frac{1}{20}\).

Key Concepts

Permutations

Permutations represent the different ways in which a set of items can be arranged or ordered. When considering permutations in probability, the order of arrangement is crucial. This distinguishes permutations from combinations, where only selection matters without regard to the arrangement order.

For example, if you have five children and you wish to determine in how many different ways they can each be assigned a unique chore over five weeks, permutations are the right tool to use. Here, each different sequence of children performing chores constitutes a unique permutation.

The formula to calculate permutations of a set is given by the factorial of the number of items, denoted as \(n!\). For five children, you calculate \(5!\), which means multiplying 5 by all positive integers less than it:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

This shows there are 120 different ways the chores can be assigned over five weeks, each representing a distinct permutation.

Case Analysis

Case analysis is an effective strategy in probability when you need to evaluate specific scenarios or conditions out of a set of possibilities. This involves breaking down a problem into individual cases to make analysis simpler and more manageable.

In the context of the problem, case analysis helps determine the number of favorable outcomes—situations which match specific criteria or conditions. Here, you evaluate the scenario where Jim is the first and Emily the last child to perform the chores.
To solve this by case analysis:

• First, identify the fixed conditions, like Jim's and Emily's specific positions in the order.
• Then, determine the outcomes with these fixed conditions. You find that there are 3 children left to fill 3 positions.
• Finally, calculate how these 3 remaining children can be arranged in those slots, using permutations of the 3 children (\(3!\)). \[3! = 3 \times 2 \times 1 = 6\]

This tells us there are 6 ways to fill the remaining positions meeting our condition.

Factorial Calculation

Factorial is a fundamental concept in mathematics and probability, used to describe the product of all positive integers up to a given number. It is denoted by an exclamation mark (\(!\)) following a number.

The factorial function is essential in problems involving permutations and combinations, as it provides the number of possible arrangements or selections in a set.

• In the problem of assigning chores to children, factorial calculation helps determine the total number of possible sequences. If you have \(n\) items to arrange, \(n!\) gives all the possible ordered combinations.
• For instance, to find how many ways 5 children can be ordered, calculate \(5!\), yielding 120. This means there are 120 unique ways to assign chores across five weeks.
• When determining specific scenarios, like Jim and Emily's assignments, factorials help calculate fewer positions by reducing the problem size. E.g., calculating for 3 remaining positions with 3 children yields \(3! = 6\).

Thus, mastering factorial calculations allows you to efficiently solve complex arrangement problems by breaking them into simpler, measurable parts.