Question

A student must complete the following sequence of courses: Two of four lab science courses, one of two literature courses, two of three mathematics courses, and one of seven physical education courses. Assume that none of these courses is a prerequisite for any other. (a) How many ways can courses be chosen if the possibility of time conflicts is disregarded? (b) How many ways can courses be chosen if two different lab courses are scheduled at the same time as one of the literature courses? (c) How many ways can courses be chosen if all the physical education courses are offered at the same time as one of the literature courses?

Short answer

(a) 252 ways; (b) 231 ways; (c) 234 ways.

Step-by-step solution

Calculate Total Ways to Choose Courses

For the lab science courses, choose 2 out of 4: \( \binom{4}{2} = 6 \). For the literature courses, choose 1 out of 2: \( \binom{2}{1} = 2 \). For the mathematics courses, choose 2 out of 3: \( \binom{3}{2} = 3 \). For the physical education courses, choose 1 out of 7: \( \binom{7}{1} = 7 \). Multiply all these together to find the total combinations: \( 6 \times 2 \times 3 \times 7 = 252 \).

Adjust for Lab and Literature Time Conflicts

Since two lab courses conflict with one literature course, we have to adjust for these situations. Select 2 labs from the remaining 2 labs (since 2 labs conflict): \( \binom{2}{2} = 1 \). Choose 1 literature course from 1 remaining: \( \binom{1}{1} = 1 \). The math and physical education selections are unaffected, so we calculate: \( 1 \times 1 \times 3 \times 7 = 21 \). Subtract these from the total: \( 252 - 21 = 231 \).

Adjust for Physical Education and Literature Time Conflicts

All physical education courses are offered at the same time as one literature course, so you can only choose the other literature course. Calculate the unaffected choices: 6 ways for labs, 3 ways for mathematics. The choices for literature and physical education conflict result in 1 combination each. Therefore, calculate: \( 6 \times 1 \times 3 \times 1 = 18 \). Subtract from the total combinations: \( 252 - 18 = 234 \).

Key concepts

Mathematical Counting

When dealing with problems that require choosing different combinations of items, like in this course selection exercise, we use mathematical counting techniques. A crucial tool here is **combinatorics**, specifically the concept of the binomial coefficient, often represented as \( \binom{n}{k} \). This notation tells us the number of ways to choose \( k \) items from a set of \( n \) items without regard to the order of selection.

In our example, for the lab science courses, choosing 2 out of 4 is calculated as \( \binom{4}{2} = 6 \). This means there are 6 different sets of 2 courses we can choose from the 4 available. Similarly, for math courses, \( \binom{3}{2} = 3 \) arises by selecting 2 out of 3 courses.

• The formula for calculating the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) denotes factorial, the product of all positive integers up to a given number.
• This kind of calculation allows us to find possible combinations efficiently, which is especially handy when dealing with large sets.

Probability in Education

Probability measures the likelihood of different outcomes, and it plays a pivotal role in educational contexts like this course selection dilemma. When choosing courses, each unique combination represents a potential outcome. The probability of each outcome is determined by the number of times it appears in comparison to the total number of possible outcomes.

In the context of our exercise, we initially found that there are 252 different ways to select the courses if there are no time conflicts. However, when potential scheduling conflicts are considered, this probability changes because fewer combinations are viable.

• Without considering time conflicts, if you choose courses randomly, each combination has a 1 in 252 probability of being selected.
• However, when two lab courses conflict with a literature class, the probability adjusts because some combinations become impossible, reducing possibilities.

Understanding probability helps students and educators make informed decisions when planning courses to optimize scheduling and avoid scheduling conflicts.

Course Selection Problem

Course selection problems often hinge on finding the optimal number of ways to fulfill certain academic requirements without creating conflicts. This requires a blend of combinatorial analysis and logical reasoning.

In this particular exercise, the problem is to ensure that the chosen courses meet the criteria from different categories like labs, math, and PE, while accounting for time conflicts. The presence of time conflicts requires that we adjust the total count of combinations. Each conflict scenario can drastically reduce the number of viable course combinations.

• For instance, if two lab courses conflict with a literature course, you cannot choose these lab courses with that particular literature option, limiting possibilities.
• When all physical education courses conflict with one literature course, you can only choose the other literature option, impacting how lab and math choices are combined.

Solving such issues provides practical insights into managing constraints and optimizing schedules, making it a valuable problem-solving exercise.