Compute The Following 8C3 5C2
Compute The Following 8C3 5C2: Your Comprehensive Guide
Welcome, fellow learner! Are you trying to compute the following 8C3 5C2 and feeling a bit overwhelmed by combination notation? Don't worry, you've come to the right place. This guide will break down this problem step-by-step, making it easy for you to understand and solve. We'll cover the basics of combinations, calculate each part individually, and then combine the results. Let's conquer this math challenge together!
Understanding Combinations: What is nCr?
Before we dive into calculating 8C3 and 5C2, let's quickly review what a combination means. In mathematics, a combination (denoted as nCr) is a way of selecting items from a larger set where the order of selection does not matter. The formula for combinations is:
nCr = n! / (r! * (n-r)!)
Here, 'n' represents the total number of items available, and 'r' represents the number of items you choose. The exclamation mark '!' denotes a factorial, meaning you multiply a number by all the positive integers less than it (e.g., 5! = 5 * 4 * 3 * 2 * 1). Understanding this formula is key to solving problems like "compute the following 8C3 5C2".
Calculating 8C3: Step-by-Step Breakdown
Now, let's apply the combination formula to find the value of 8C3. In this specific instance, n = 8 and r = 3. We will break this down into clear, manageable steps.
First, calculate the necessary factorials:
- 8! (8 factorial) = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
- 3! (3 factorial) = 3 × 2 × 1 = 6
- (8-3)! = 5! (5 factorial) = 5 × 4 × 3 × 2 × 1 = 120
Next, substitute these values into the combination formula:
8C3 = 8! / (3! × (8-3)!) = 8! / (3! × 5!)Finally, perform the division to get your result:
8C3 = 40,320 / (6 × 120) = 40,320 / 720 = 56
So, 8C3 equals 56. This means there are 56 distinct ways to choose 3 items from a set of 8, where the order of selection does not matter.
Calculating 5C2: Breaking Down the Second Part
Great job with 8C3! Let's move on to the second part of our problem, 5C2. Here, n = 5 and r = 2. We'll follow a similar process to find its value.
Determine the required factorials for this part:
- 5! (5 factorial) = 5 × 4 × 3 × 2 × 1 = 120
- 2! (2 factorial) = 2 × 1 = 2
- (5-2)! = 3! (3 factorial) = 3 × 2 × 1 = 6
Plug these factorial values into the combination formula:
5C2 = 5! / (2! × (5-2)!) = 5! / (2! × 3!)Execute the division to find the final result for 5C2:
5C2 = 120 / (2 × 6) = 120 / 12 = 10
Thus, 5C2 equals 10. This indicates there are 10 unique ways to choose 2 items from a group of 5 without considering the order of selection.
Putting It All Together: Compute 8C3 × 5C2
You've successfully calculated both individual combinations! Now, to fully compute the following 8C3 5C2, we simply need to multiply our two results together.
We found that:
- 8C3 = 56
- 5C2 = 10
Therefore, the final calculation for the entire expression is:
8C3 × 5C2 = 56 × 10 = 560
The answer to compute the following 8C3 5C2 is 560. Fantastic work! You've mastered a complex combination problem.
Conclusion
In this comprehensive guide, we've walked through the entire process to compute the following 8C3 5C2. We started by understanding the fundamental formula for combinations, which is crucial for these types of problems. Then, we systematically calculated 8C3, arriving at 56, and subsequently calculated 5C2, which yielded 10. By multiplying these individual results, we confidently arrived at the final answer of 560. Mastering combinations is a valuable skill in probability, statistics, and many real-world applications, and you've just taken a significant step in that direction. Keep practicing, and these types of problems will become second nature!
Frequently Asked Questions (FAQ)
- What is the key difference between a combination and a permutation?
The main difference lies in order. In combinations, the order of selection does not matter (e.g., choosing 3 friends for a team is the same regardless of who you pick first). In permutations, the order does matter (e.g., arranging 3 friends in specific seats).
- Why is understanding factorials so important when working with combinations?
Factorials are fundamental because they represent the total number of ways to arrange a set of distinct items. They form the core building blocks for both combination and permutation formulas, simplifying complex counting problems.
- Can I use a calculator to compute combinations like 8C3 or 5C2?
Yes, absolutely! Most scientific and graphing calculators have an 'nCr' button (often found under a 'Prob' or 'Math' menu) that can quickly compute combinations. While calculators are convenient, understanding the manual calculation process is essential for grasping the underlying mathematical concept.
- What are some practical, real-world applications of combinations?
Combinations are used in various fields. Examples include calculating probabilities in card games or lotteries, determining the number of possible teams or committees that can be formed from a group, understanding genetic possibilities, and even in computer science for algorithm design and data analysis.
Compute The Following 8C3 5C2
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