Understanding Combinations: Solving C(8, 4) ÷ C(5, 3) and Similar Problems

Welcome to our guide on how to solve combinatorial problems, specifically focusing on the division of combinations. We'll explore the concept of combinations and how they are used in mathematics, particularly in solving problems like C(8, 4) ÷ C(5, 3). By the end of this article, you will have a clear understanding of how to approach and solve similar problems with ease.

What Are Combinations?

Combinations are a fundamental concept in combinatorial mathematics and probability theory. A combination is a selection of items from a larger set, without regard to the order of selection. The formula for calculating the number of combinations is given by:

Formula for Combinations

Combinations, denoted as C(n, r) or , represent the number of ways to choose r items from a set of n items. The formula is:

C(n, r) frac{n!}{r!(n-r)!}

Where n! denotes the factorial of n, which is the product of all positive integers up to n. For instance, 5! 5 × 4 × 3 × 2 × 1.

Solving C(8, 4) ÷ C(5, 3)

Let's break down the problem step by step to illustrate how to solve C(8, 4) ÷ C(5, 3).

Step 1: Calculate C(8, 4)

To find C(8, 4), we use the combinations formula:

C(8, 4) frac{8!}{4!(8-4)!} frac{8!}{4!4!}

Now, let's compute the factorials step by step:

frac{8 times 7 times 6 times 5 times 4!}{4! times 4!}

Notice that the 4! in the numerator and denominator cancel out:

8 times 7 times 6 times 5 1680 ÷ 24 70

Step 2: Calculate C(5, 3)

Next, we calculate C(5, 3):

C(5, 3) frac{5!}{3!(5-3)!} frac{5!}{3!2!}

Again, we compute the factorials step by step:

frac{5 times 4 times 3!}{3! times 2!}

The 3! in the numerator and denominator cancel out:

frac{5 times 4}{2} 10

Step 3: Perform the Division

Now that we have both values, we can perform the division:

C(8, 4) ÷ C(5, 3) 70 ÷ 10 7

Beyond C(8, 4) ÷ C(5, 3)

While our primary focus is to understand how to solve problems like C(8, 4) ÷ C(5, 3), it's valuable to explore more complex scenarios. Here are some additional examples to help you further refine your problem-solving skills:

Example 1: C(10, 5) ÷ C(7, 3)

Let's break it down:

Step 1: Calculate C(10, 5)

C(10, 5) frac{10!}{5!(10-5)!} frac{10!}{5!5!} frac{10 times 9 times 8 times 7 times 6}{5 times 4 times 3 times 2 times 1} 252

Step 2: Calculate C(7, 3)

C(7, 3) frac{7!}{3!(7-3)!} frac{7!}{3!4!} frac{7 times 6 times 5}{3 times 2 times 1} 35

Step 3: Perform the Division

C(10, 5) ÷ C(7, 3) 252 ÷ 35 7.2

Example 2: C(12, 6) ÷ C(9, 4)

Similarly, let's break it down:

Step 1: Calculate C(12, 6)

C(12, 6) frac{12!}{6!(12-6)!} frac{12!}{6!6!} frac{12 times 11 times 10 times 9 times 8 times 7}{6 times 5 times 4 times 3 times 2 times 1} 924

Step 2: Calculate C(9, 4)

C(9, 4) frac{9!}{4!(9-4)!} frac{9!}{4!5!} frac{9 times 8 times 7 times 6}{4 times 3 times 2 times 1} 126

Step 3: Perform the Division

C(12, 6) ÷ C(9, 4) 924 ÷ 126 7.333333333333333

Conclusion

Understanding and solving problems involving combinations is a crucial skill in mathematics, particularly when dealing with probability and statistics. By mastering the formula and step-by-step approach, you can confidently tackle a wide range of problems. Whether you're studying for an exam or simply interested in enhancing your problem-solving skills, practice will deepen your understanding.