Problem 73 - College Algebra with Modeling and Visualization

Question

How many ways are there to draw 3 red marbles and 2 blue marbles from a jar that contains 10 red marbles and 12 blue marbles?

Step-by-Step Solution

Answer

There are 7920 ways to draw the marbles.

Step 1: Understand the Problem

We need to find how many ways there are to draw a specific combination of marbles from a jar. We're drawing 3 red marbles from a total of 10 red marbles and 2 blue marbles from a total of 12 blue marbles.

Step 2: Combination Formula

The problem involves using the combination formula because the order of selection does not matter. The combination formula is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Calculate Red Marble Combinations

We are choosing 3 red marbles from 10. Plug these into the combination formula: \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \).

Step 4: Calculate Blue Marble Combinations

Next, we choose 2 blue marbles from 12 blue marbles. Plug these into the combination formula: \( \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \).

Step 5: Total Number of Ways

Since the selections of red and blue marbles are independent, the total number of ways to draw the marbles is the product of the combinations found in the previous steps: \( 120 \times 66 = 7920 \).

Key Concepts

Understanding the Combination Formula

The combination formula is a vital concept in combinatorics, helping us determine how many ways we can select items from a larger pool, without worrying about the order of selection. When faced with a problem like selecting marbles of different colors from a jar, the order in which we pick marbles doesn’t matter—it's the specific number we pick that's crucial. The combination formula is expressed as:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here's a breakdown:

• ! means "n factorial", multiplying all numbers from 1 to n together.
• "n" represents the total number of items you're choosing from.
• "r" is the number of items to select.

By using the combination formula, we calculate how many different sets of marbles we can pick. For example, choosing 3 red marbles from 10 means using the combination formula \( \binom{10}{3} \). And this gives us 120 different possible sets of red marbles. Similarly, finding combinations for blue marbles lets us calculate countless color assortments efficiently.

Probability and Its Role in Combinations

Probability is about finding the chances of a specific event happening. And in our context of selecting marbles, it aligns perfectly with the idea of combinations. Whether for games, experiments, or simple fun jar-picking, probability tells us the likelihood of a particular draw occurring exactly the way we want. To understand how probability plays with combination, imagine this jar again. We want to know how likely we are to draw a particular set of marbles by chance. This would imply knowing the ratio of our successful draws (our combinations) to all possible draws. However, in this exercise, we aren't asked to calculate probability but instead focus purely on knowing the number of favorable outcomes. But it’s interesting to note how probability would interact with combinations to inform us of these chances. When we compute the combinations both for red and blue marbles and multiply them—as seen in the exercise—we basically look at all possible successful draws. The understanding of these draws is deeply rooted in probability. Knowing probability lays the groundwork for when we do want to calculate actual odds in future exercises or real-world applications.

Binomial Coefficient: The Heart of Combination Problems

The binomial coefficient is denoted by \( \binom{n}{r} \) and is often synonymous with the combination formula. It's nicknamed so because it appears prominently in binomial expansions—a separate topic rich in its own practical mathematical uses.In our scenario with marble picking: \( \binom{10}{3} \) and \( \binom{12}{2} \) are essentially binomial coefficients, highlighting their importance in choosing marbles.These coefficients serve other roles beyond our example. They're key in:

• Polynomial expansions: helping to expand expressions like \((x+y)^n\).
• Probability theory: as explored previously.
• Simplifying complex problems that involve numerous combinations.

Why are binomial coefficients often brought up? Because they simplify how we understand the number of ways items can be grouped, not just in probability but also across various realms of mathematics.