Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. If your college offers 400 courses, 20 of which are in mathematics, and your counselor arranges your schedule of 4 courses by random selection, how many schedules are possible that do not include a math course?

Step-by-Step Solution

Answer

There are 21,752,630 possible schedules without a math course.

Step 1: Understand the Problem

We need to find the number of possible schedules that do not include a math course out of a total of 400 courses.

Step 2: Identify Total Courses Excluding Math

Since there are 20 math courses, we subtract these from the total courses, resulting in 380 non-math courses, because 400 - 20 = 380.

Step 3: Select Courses Randomly

To form a schedule, we select 4 courses that do not involve any math courses. Since there are 380 non-math courses, we need to use the combinations formula to select 4 courses.

Step 4: Apply Combinations Formula

The number of ways to choose 4 courses from 380 total without regard to order is calculated using the combinations formula:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \(n = 380\) and \(r = 4\).

Step 5: Calculate the Combination

Substitute 380 for \(n\) and 4 for \(r\) into the combinations formula:\[\binom{380}{4} = \frac{380!}{4!(380-4)!} = \frac{380 \times 379 \times 378 \times 377}{4 \times 3 \times 2 \times 1}\]This calculation simplifies to a large number.

Step 6: Perform the Calculation

After performing the multiplication and division:\[\binom{380}{4} = 21,752,630\]

Key Concepts

Fundamental Principle of Counting

The Fundamental Principle of Counting is like a blueprint for solving problems that involve figuring out how many ways a sequence of events can happen. It's simple yet powerful, as it allows us to calculate the total number of outcomes by multiplying the number of ways each event can occur. For instance, if there are 3 shirts and 4 pants, and you want to know how many outfit combinations you can make, you multiply 3 by 4, getting 12 possible outfits. This principle applies when each event in a sequence is independent, meaning the outcome of one does not affect the others.

• Identify the sequence of events.
• Determine how many ways each event can occur.
• Multiply the numbers to get the total number of combinations.

This principle is foundational in combinatorics because it provides a straightforward way to compute complex scenarios when combined with other principles and rules.

Permutations

Permutations are all about arrangements. When the order of items matters, we use permutations to determine how many distinct arrangements we can form. Imagine having three different books and wanting to place them on a shelf. With permutations, you consider every possible order they can be arranged in.
The formula for permutations when arranging all items is:\[ \text{n!} \]Where \( n \) is the number of items to arrange. Each term in this formula reduces by one as you move to the next position, creating a factorial.

When Order Matters

• Useful in races: Who gets 1st, 2nd, 3rd?
• Ordering in line: Which person stands where?

If only some items are chosen out of a larger set:\[ \frac{n!}{(n-r)!} \]This formula reflects choosing \( r \) items from \( n \) available, showing how permutations allow us to deeply consider order and detail in arrangements.

Combinations

Combinations are like the flip side of permutations. With combinations, the order of items does not matter. This is especially important when you need to choose items from a set, and the sequence is irrelevant. For example, picking four friends to form a team doesn't rely on who is chosen first or last.
The combinations formula is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n \) represents the total number of items, and \( r \) the number of items to choose.

When Order Doesn't Matter

• Drawing a hand of cards: Only the final hand matters, not the order of the draw.
• Creating groups: Forming a committee or a sports team.

In our earlier problem, combinations helped us find how many ways we can select 4 courses from 380 without including math, offering insight into the versatility of combinations in various scenarios.