Not considering the order of selection

Oct 13, 2025 by thesweetdesigns.com 136 views

Introduction

In mathematics, combinations are a fundamental concept used to calculate the number of ways to choose a certain number of items from a larger set, without regard to the order of selection. The formula for calculating combinations is given by:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

In this article, we will compute the value of the expression ${_8 C_3 \cdot _5 C_2}$ and determine the correct answer from the given options.

Understanding the Problem

The given expression involves two combinations:

${_8 C_3}$ : This represents the number of ways to choose 3 items from a set of 8.
${_5 C_2}$ : This represents the number of ways to choose 2 items from a set of 5.

We need to compute the product of these two combinations.

Computing ${_8 C_3}$

To compute ${_8 C_3}$ , we can use the formula for combinations:

C(n, k) = n! / (k!(n-k)!)

In this case, n = 8 and k = 3. Plugging these values into the formula, we get:

C(8, 3) = 8! / (3!(8-3)!) C(8, 3) = 8! / (3!5!) C(8, 3) = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(5 × 4 × 3 × 2 × 1)) C(8, 3) = 56

So, ${_8 C_3}$ = 56.

Computing ${_5 C_2}$

To compute ${_5 C_2}$ , we can use the formula for combinations:

C(n, k) = n! / (k!(n-k)!)

In this case, n = 5 and k = 2. Plugging these values into the formula, we get:

C(5, 2) = 5! / (2!(5-2)!) C(5, 2) = 5! / (2!3!) C(5, 2) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) C(5, 2) = 10

So, ${_5 C_2}$ = 10.

Computing the Product

Now that we have computed the values of ${_8 C_3}$ and ${_5 C_2}$ , we can compute the product:

${_8 C_3 \cdot _5 C_2}$ = 56 × 10 ${_8 C_3 \cdot _5 C_2}$ = 560

Therefore, the correct answer is:

D. 560

Conclusion

Introduction

In our previous article, we explored the concept of combinations and computed the value of the expression ${_8 C_3 \cdot _5 C_2}$ . In this article, we will address some frequently asked questions related to combinations and permutations.

Q: What is the difference between combinations and permutations?

A: Combinations and permutations are both used to calculate the number of ways to choose a certain number of items from a larger set. However, the key difference between the two is that permutations take into account the order of selection, while combinations do not.

Q: How do I calculate the number of permutations?

A: To calculate the number of permutations, you can use the formula:

P(n, k) = n! / (n-k)!

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

Q: What is the formula for combinations?

A: The formula for combinations is:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

Q: How do I calculate the number of combinations when n and k are large?

A: When n and k are large, it can be difficult to calculate the number of combinations using the formula. In such cases, you can use the following approximation:

C(n, k) ≈ (n^k) / (k! * e^k)

where e is the base of the natural logarithm.

Q: What is the relationship between combinations and binomial coefficients?

A: Combinations and binomial coefficients are closely related. The binomial coefficient ${_n C_k}$ is equal to the number of combinations of n items taken k at a time.

Q: How do I use combinations in real-world applications?

A: Combinations have numerous applications in real-world scenarios, such as:

Statistics: Combinations are used to calculate the number of ways to choose a sample from a population.
Computer Science: Combinations are used in algorithms for solving problems such as the traveling salesman problem.
Finance: Combinations are used to calculate the number of ways to choose a portfolio of assets.

Q: What are some common mistakes to avoid when working with combinations?

A: Some common mistakes to avoid when working with combinations include:

Not considering the order of selection: Combinations do not take into account the order of selection, so make sure to use the correct formula.
Not using the correct formula: Make sure to use the correct formula for combinations, which is C(n, k) = n! / (k!(n-k)!).
Not considering the value of k: Make sure to consider the value of k when calculating combinations, as it can affect the result.

Conclusion

In this article, we addressed some frequently asked questions related to combinations and permutations. We covered topics such as the difference between combinations and permutations, how to calculate the number of permutations, and how to use combinations in real-world applications. We also highlighted some common mistakes to avoid when working with combinations. By understanding these concepts, you can better apply combinations and permutations in your own work.