Compute The Following: 8 C 8 {}_8C_8 8 ​ C 8 ​ A. 16 B. 0 C. 64 D. 1

Introduction

Binomial coefficients are a fundamental concept in mathematics, particularly in combinatorics and probability theory. They are 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. In this article, we will focus on computing the binomial coefficient ${}_8C_8$, which is a specific type of binomial coefficient.

What is a Binomial Coefficient?

A binomial coefficient, often referred to as "n choose k", is a mathematical expression that represents the number of ways to choose k items from a set of n items, without regard to the order of selection. It is denoted by the symbol ${}_nC_k$ or $\binom{n}{k}$. The formula for calculating a binomial coefficient is:

${}_nC_k = \frac{n!}{k!(n-k)!}$

where n! represents the factorial of n, which is the product of all positive integers from 1 to n.

Computing ${}_8C_8$

To compute ${}_8C_8$, we can use the formula above. We need to calculate the factorial of 8, as well as the factorials of 8 and 0.

Calculating Factorials

The factorial of a number n, denoted by n!, is the product of all positive integers from 1 to n. For example, the factorial of 5 is:

5! = 5 × 4 × 3 × 2 × 1 = 120

We can calculate the factorial of 8 as follows:

8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320

Computing ${}_8C_8$

Now that we have calculated the factorial of 8, we can compute ${}_8C_8$ using the formula above:

${}_8C_8 = \frac{8!}{8!(8-8)!} = \frac{40320}{40320 \times 1} = 1$

Conclusion

In this article, we have computed the binomial coefficient ${}_8C_8$ using the formula for binomial coefficients. We have also discussed the concept of binomial coefficients and how they are used in mathematics. The result of our computation is ${}_8C_8 = 1$, which is the correct answer.

Why is ${}_8C_8 = 1$?

The reason why ${}_8C_8 = 1$ is because there is only one way to choose 8 items from a set of 8 items, which is to choose all 8 items. This is because there are no other items to choose from, so the only possible choice is to choose all 8 items.

Real-World Applications

Binomial coefficients have many real-world applications, including:

• Probability theory: Binomial coefficients are used to calculate the probability of certain events occurring.
• Combinatorics: Binomial coefficients are used to count the number of ways to arrange objects in a certain order.
• Statistics: Binomial coefficients are used to calculate the probability of certain events occurring in a sample of data.

Common Mistakes

When computing binomial coefficients, there are several common mistakes to avoid:

• Not using the correct formula: Make sure to use the correct formula for calculating binomial coefficients.
• Not calculating factorials correctly: Make sure to calculate factorials correctly, as small errors can lead to large mistakes.
• Not simplifying expressions: Make sure to simplify expressions as much as possible to avoid unnecessary calculations.

Conclusion

Q: What is a binomial coefficient?

A: A binomial coefficient, often referred to as "n choose k", is a mathematical expression that represents the number of ways to choose k items from a set of n items, without regard to the order of selection.

Q: How do I calculate a binomial coefficient?

A: To calculate a binomial coefficient, you can use the formula:

${}_nC_k = \frac{n!}{k!(n-k)!}$

where n! represents the factorial of n, which is the product of all positive integers from 1 to n.

Q: What is the difference between ${}_nC_k$ and ${}_kC_n$?

A: ${}_nC_k$ and ${}_kC_n$ are two different binomial coefficients. ${}_nC_k$ represents the number of ways to choose k items from a set of n items, while ${}_kC_n$ represents the number of ways to choose n items from a set of k items. They are not the same, but they are related by the formula:

${}_kC_n = \frac{{}_nC_k}{(n-k)!}$

Q: How do I calculate the factorial of a number?

A: To calculate the factorial of a number n, you can use the formula:

n! = n × (n-1) × (n-2) × ... × 2 × 1

For example, the factorial of 5 is:

5! = 5 × 4 × 3 × 2 × 1 = 120

Q: What is the relationship between binomial coefficients and Pascal's triangle?

A: Binomial coefficients are closely related to Pascal's triangle. Each entry in Pascal's triangle is a binomial coefficient. The formula for calculating binomial coefficients can be used to generate Pascal's triangle.

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

A: Binomial coefficients have many real-world applications, including:

• Probability theory: Binomial coefficients are used to calculate the probability of certain events occurring.
• Combinatorics: Binomial coefficients are used to count the number of ways to arrange objects in a certain order.
• Statistics: Binomial coefficients are used to calculate the probability of certain events occurring in a sample of data.

Q: What are some common mistakes to avoid when computing binomial coefficients?

A: Some common mistakes to avoid when computing binomial coefficients include:

• Not using the correct formula: Make sure to use the correct formula for calculating binomial coefficients.
• Not calculating factorials correctly: Make sure to calculate factorials correctly, as small errors can lead to large mistakes.
• Not simplifying expressions: Make sure to simplify expressions as much as possible to avoid unnecessary calculations.

Q: Can I use a calculator or computer program to compute binomial coefficients?

A: Yes, you can use a calculator or computer program to compute binomial coefficients. Many calculators and computer programs have built-in functions for calculating binomial coefficients.

Q: How do I prove that ${}_nC_k = \frac{n!}{k!(n-k)!}$?

A: To prove that ${}_nC_k = \frac{n!}{k!(n-k)!}$, you can use a combinatorial argument. Consider a set of n items, and choose k items from it. The number of ways to do this is equal to the number of ways to choose k items from a set of n items, which is given by the formula:

${}_nC_k = \frac{n!}{k!(n-k)!}$

This formula can be proved using a variety of methods, including combinatorial arguments and algebraic manipulations.

Conclusion

In conclusion, binomial coefficients are a fundamental concept in mathematics that have many real-world applications. By understanding how to compute binomial coefficients, we can solve a wide range of problems in probability theory, combinatorics, and statistics.