Calculating Combinations: Marble Selection Scenarios

Hey guys, let's dive into a classic probability problem! We've got a box overflowing with colorful marbles – 5 blue and 6 red ones. The big question is: How many different ways can we pull out 3 marbles, based on specific color combinations? This kind of problem is all about combinations, where the order of the marbles doesn't matter. Getting a blue, then a red, then another blue is the same as getting a red, then a blue, then a blue – it's the final group that counts. So, let's break down each scenario step-by-step to see how many unique combinations we can create. We'll use some basic combinatorics principles to get our answers. Buckle up, it's gonna be fun!

a. All Marbles are Blue

Alright, first up, we're focusing on the scenario where we grab 3 blue marbles. Since we have 5 blue marbles in the box, we need to figure out how many ways we can choose 3 out of those 5. This is a combination problem, and the formula for combinations is: nCr = n! / (r! * (n-r)!), where:

• n is the total number of items to choose from (in our case, the total number of blue marbles).
• r is the number of items we're choosing (in our case, the number of blue marbles we want to pick).
• ! means factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).

So, for our problem, n = 5 (total blue marbles) and r = 3 (marbles we want to pick). Let's plug it into the formula:

5C3 = 5! / (3! * (5-3)!) = 5! / (3! * 2!)

Now, let's calculate those factorials:

5! = 5 * 4 * 3 * 2 * 1 = 120 3! = 3 * 2 * 1 = 6 2! = 2 * 1 = 2

Substitute the factorials back into the equation:

5C3 = 120 / (6 * 2) = 120 / 12 = 10

So, there are 10 different ways to select 3 blue marbles from the box. That's our answer for scenario a! Pretty straightforward, right? We've effectively used the combination formula to pinpoint the number of unique groupings possible when choosing only blue marbles. The key here is understanding that the order in which you pick the marbles doesn't matter; only the final combination matters.

To recap, the formula allowed us to efficiently calculate the number of possible combinations without manually listing each one. This method becomes especially useful as the numbers of marbles increase, making it easier to find the possible scenarios in similar problems.

b. 2 Red Marbles and 1 Blue Marble

Next up, we want to find out how many ways we can pick a group with 2 red marbles and 1 blue marble. This is a bit more complex because we're dealing with two different colors, so we'll need to break it down into two separate combination calculations and then combine them. Let's do this in steps:

First, we'll calculate the number of ways to choose 2 red marbles from the 6 available. Again, we use the combination formula: nCr = n! / (r! * (n-r)!), but this time:

• n = 6 (total red marbles)
• r = 2 (red marbles we want to pick)

So, 6C2 = 6! / (2! * (6-2)!) = 6! / (2! * 4!)

Let's calculate those factorials:

6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 2! = 2 * 1 = 2 4! = 4 * 3 * 2 * 1 = 24

Substitute the factorials back into the equation:

6C2 = 720 / (2 * 24) = 720 / 48 = 15

There are 15 ways to choose 2 red marbles. Great, now we handle the blue marble. We need to pick 1 blue marble from the 5 available. The combination formula becomes:

5C1 = 5! / (1! * (5-1)!) = 5! / (1! * 4!)

5! = 5 * 4 * 3 * 2 * 1 = 120 1! = 1 4! = 4 * 3 * 2 * 1 = 24

5C1 = 120 / (1 * 24) = 120 / 24 = 5

There are 5 ways to choose 1 blue marble. Now, since each selection of red marbles can be combined with each selection of blue marbles, we need to multiply the two results together. The total number of ways to get 2 red and 1 blue is 15 (ways to choose red) * 5 (ways to choose blue) = 75. So, there are 75 different ways to pick 2 red marbles and 1 blue marble.

This scenario highlights a key principle: when independent events must both happen to achieve a desired outcome, we multiply the number of ways each event can occur. This process of breaking down the problem into smaller parts and combining the results allows us to solve complex combinations problems methodically. It’s all about breaking down the problem and applying the right formula in each step to reach a solution.

Conclusion: Mastering Marble Combinations

So, there you have it, guys! We've successfully navigated the colorful world of marble combinations. We've seen how to calculate the number of ways to select marbles when we have specific color requirements. Remember, the combination formula, nCr = n! / (r! * (n-r)!), is your best friend for these kinds of problems. Being able to apply this is key to solving different types of scenarios.

• For picking only blue marbles, we directly applied the formula.
• For the mixed color scenario, we broke the problem down into smaller parts, solved each separately using the combination formula, and then combined the results through multiplication.

The key takeaway? Breaking down complex problems into smaller, manageable parts and using the appropriate formula for each part. Practicing these problems helps solidify your understanding of combinations and probability, and shows how the concepts can be applied to real-world scenarios. Keep practicing these concepts, and you'll be a combinations pro in no time! Feel free to try variations of these problems – change the number of marbles, or the colors, to test your understanding. The more you practice, the better you'll get!

Keep experimenting, and keep having fun with math! Understanding how to calculate combinations is a fundamental skill that applies far beyond just marbles. It's useful in fields like statistics, data analysis, and even everyday decision-making.