Problem 51 - Algebra 2 and Trigonometry

Question

A local convenience store hires three students to work after school. Next month, there are 20 days on which they will work. Alex will work 8 days, Rosa will work 6 days, and Carla will work 6 days. In how many ways can their schedule for the month be arranged?

Step-by-Step Solution

Answer

The schedule can be arranged in 50,388,378 ways.

Step 1: Calculate Total Days Worked

First, calculate the total number of schedule arrangements possible for Alex, Rosa, and Carla. Since Alex, Rosa, and Carla are all working on 20 days combined, we need to distribute these 20 days among them: Alex works 8 days, Rosa works 6 days, and Carla works 6 days.

Step 2: Apply Multinomial Theorem

The number of arrangements can be found using the multinomial coefficient. This is because we are dividing the total 20 days into three groups of differing amounts (8, 6, and 6). The general formula for a multinomial distribution is: \[ \frac{n!}{k_1!k_2!...k_m!} \]where \( n \) is the total number of items to arrange, and \( k_1, k_2, \, ... \, k_m \) are the sizes of each group.

Step 3: Plug Values into the Formula

Plug the values into the formula:\[\frac{20!}{8!6!6!} \]Here, 20 is the total days to arrange, 8 is the number of days Alex works, and 6 each for Rosa and Carla.

Step 4: Calculate Factorials

Calculate the factorials: - \( 20! = 20 \times 19 \times 18 \times ... \times 1 \)- \( 8! = 8 \times 7 \times ... \times 1 \)- \( 6! = 6 \times 5 \times ... \times 1 \)

Step 5: Simplify and Calculate the Multinomial Coefficient

Now, simplify and calculate the expression:\[20! = 2,432,902,008,176,640,000 \8! = 40,320 \6! = 720 \\]Substituting these into the multinomial formula gives:\[\frac{2,432,902,008,176,640,000}{40,320 \times 720 \times 720} \approx 50,388,378 \]

Step 6: Interpret the Result

The result, 50,388,378, represents the number of distinct ways the work schedule for Alex, Rosa, and Carla can be arranged over the 20 days.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It helps us find out how many ways we can organize a set of items, like creating a schedule for workers.

In scheduling problems, combinatorics is used to determine the possible arrangements by considering all the workers and the days they can work. In our example, if we have 20 working days for the store, and we need to distribute these days among three workers, we use combinatorics to find the total number of ways this can be done.

• We assign a specific number of days to each worker (8 for Alex, 6 for Rosa, and 6 for Carla).
• The objective is to see how these days can be assigned without any overlaps, meaning each arrangement is unique.
• This arrangement process often involves solving with combinations and permutations, which are critical tools in combinatorics.

Although it might sound complex, combinatorics essentially boils down to counting all potential outcomes while respecting the constraints placed on the problem, like the number of days each person works.

Factorials

Factorials are incredibly useful in combinatorics, especially when dealing with problems that involve counting arrangements and schedules. A factorial, represented by an exclamation mark (!), multiplies a series of descending natural numbers. For instance, 4 factorial (written as 4!) equals 4 x 3 x 2 x 1.

In the context of our example, where we need to arrange 20 workdays among Alex, Rosa, and Carla, factorials come into play when calculating the number of possible arrangements of these days. Factorials allow us to compute permutations and combinations efficiently by providing a quick way to represent repeated multiplication of numbers.

• For Alex, with 8 working days, we calculate 8! (8 factorial).
• For Rosa and Carla, who each work 6 days, we need to calculate 6! for their days.
• The complete arrangement is then calculated using the multinomial formula, which uses these factorials to explore all possible scenarios.

Despite seeming tedious at first, understanding factorials makes it easier to tackle even the most complex combinatorial problems by allowing large calculations to be broken down into manageable parts.

Scheduling problems

Scheduling problems are a part of everyday life issues where combinatorics and factorials come into play to find the best or most feasible arrangement of tasks over time. They arise in various scenarios, not only in work environments but in any situation where resources need to be allocated efficiently.

In our problem, scheduling means determining how to allocate 20 total working days among Alex, Rosa, and Carla in the most abundant number of unique ways. We aim to consider all potential permutations, leaving no overlap and satisfying the conditions for each worker's availability and required workdays.

• The challenge lies in distributing workdays such that Alex works precisely 8 days and Rosa and Carla work 6 days each.
• We apply mathematical principles that leverage factorials and the multinomial theorem to ensure all possible unique schedules are considered.
• Such problems are frequently encountered in operations research, project management, and resource allocation fields, highlighting the importance of mathematical solutions in planning and organizing real-world tasks.

Mastering scheduling problems involves understanding how to apply mathematical concepts like the multinomial theorem in practical situations, ensuring that tasks and resources are optimally assigned within given constraints.