Evaluate The Expression: $\binom{47}{46}$

Feb 26, 2025 by ADMIN 44 views

$Evaluate the Expression: $\binom{47}{46}$$

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Introduction

In mathematics, the binomial coefficient, often referred to as "n choose k", is a way to calculate the number of combinations of a set of items taken k at a time without regard to the order of selection. It is denoted by the symbol $\binom{n}{k}$ and is calculated as $\frac{n!}{k!(n-k)!}$ . In this article, we will evaluate the expression $\binom{47}{46}$ .

Understanding the Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics and probability theory. It is used to calculate the number of ways to choose k items from a set of n items, where the order of selection does not matter. The formula for the binomial coefficient is:

\binom{n}{k} = \frac{n!}{k!(n-k)!}

where $n!$ denotes the factorial of n, which is the product of all positive integers less than or equal to n.

Evaluating the Expression

To evaluate the expression $\binom{47}{46}$ , we can use the formula for the binomial coefficient:

\binom{47}{46} = \frac{47!}{46!(47-46)!}

Simplifying the expression, we get:

\binom{47}{46} = \frac{47!}{46!1!}

\binom{47}{46} = \frac{47 \times 46!}{46!}

\binom{47}{46} = 47

Conclusion

In conclusion, the expression $\binom{47}{46}$ evaluates to 47. This is because the binomial coefficient is used to calculate the number of ways to choose k items from a set of n items, and in this case, we are choosing 46 items from a set of 47 items, which is equivalent to choosing 1 item from a set of 47 items.

Real-World Applications

The binomial coefficient has many real-world applications, including:

Probability theory: The binomial coefficient is used to calculate the probability of certain events occurring.
Combinatorics: The binomial coefficient is used to calculate the number of ways to arrange objects in a particular order.
Statistics: The binomial coefficient is used to calculate the probability of certain events occurring in a statistical sample.

Example Use Cases

Here are a few example use cases for the binomial coefficient:

Choosing a team: Suppose we have a team of 47 players and we want to choose a team of 46 players. We can use the binomial coefficient to calculate the number of ways to choose the team.
Calculating probabilities: Suppose we have a fair coin and we want to calculate the probability of getting exactly 46 heads in 47 flips. We can use the binomial coefficient to calculate the probability.

Code Implementation

Here is an example implementation of the binomial coefficient in Python:

import math

def binomial_coefficient(n, k):
    return math.comb(n, k)

print(binomial_coefficient(47, 46))

This code uses the math.comb function to calculate the binomial coefficient.

Conclusion

In conclusion, the expression $\binom{47}{46}$ evaluates to 47. The binomial coefficient is a fundamental concept in combinatorics and probability theory, and it has many real-world applications. We have also provided an example implementation of the binomial coefficient in Python.

References

Wikipedia: Binomial coefficient. Retrieved from https://en.wikipedia.org/wiki/Binomial_coefficient
Khan Academy: Binomial coefficient. Retrieved from https://www.khanacademy.org/math/combinatorics-combinatorics-arrangements/binomial-coefficient/v/binomial-coefficient

Frequently Asked Questions

Q: What is the binomial coefficient?

A: The binomial coefficient, often referred to as "n choose k", is a way to calculate the number of combinations of a set of items taken k at a time without regard to the order of selection.

Q: How is the binomial coefficient calculated?

A: The binomial coefficient is calculated using the formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ , where $n!$ denotes the factorial of n, which is the product of all positive integers less than or equal to n.

Q: What is the difference between the binomial coefficient and the factorial?

A: The binomial coefficient is a way to calculate the number of combinations of a set of items taken k at a time, while the factorial is a way to calculate the product of all positive integers less than or equal to n.

Q: Can you provide an example of how to use the binomial coefficient?

A: Suppose we have a team of 47 players and we want to choose a team of 46 players. We can use the binomial coefficient to calculate the number of ways to choose the team. The number of ways to choose 46 players from a team of 47 players is given by $\binom{47}{46}$ .

Q: How do you calculate the binomial coefficient for large values of n?

A: For large values of n, it is more efficient to use the formula: $\binom{n}{k} = \frac{n \times (n-1) \times (n-2) \times \ldots \times (n-k+1)}{k!}$ , rather than calculating the factorial of n.

Q: Can you provide a Python implementation of the binomial coefficient?

A: Yes, here is an example implementation of the binomial coefficient in Python:

import math

def binomial_coefficient(n, k):
    return math.comb(n, k)

print(binomial_coefficient(47, 46))

Q: What are some real-world applications of the binomial coefficient?

A: The binomial coefficient has many real-world applications, including:

Probability theory: The binomial coefficient is used to calculate the probability of certain events occurring.
Combinatorics: The binomial coefficient is used to calculate the number of ways to arrange objects in a particular order.
Statistics: The binomial coefficient is used to calculate the probability of certain events occurring in a statistical sample.

Q: Can you provide some example use cases for the binomial coefficient?

A: Here are a few example use cases for the binomial coefficient:

Choosing a team: Suppose we have a team of 47 players and we want to choose a team of 46 players. We can use the binomial coefficient to calculate the number of ways to choose the team.
Calculating probabilities: Suppose we have a fair coin and we want to calculate the probability of getting exactly 46 heads in 47 flips. We can use the binomial coefficient to calculate the probability.

Common Mistakes

Calculating the factorial of n instead of the binomial coefficient: This can lead to incorrect results, especially for large values of n.
Not using the correct formula for the binomial coefficient: This can lead to incorrect results, especially for large values of n.
Not considering the order of selection: This can lead to incorrect results, especially when calculating the number of ways to arrange objects in a particular order.

Conclusion

In conclusion, the binomial coefficient is a fundamental concept in combinatorics and probability theory, and it has many real-world applications. We have also provided an example implementation of the binomial coefficient in Python and discussed some common mistakes to avoid.

Evaluate The Expression: $\binom{47}{46}$

Introduction

Understanding the Binomial Coefficient

Evaluating the Expression

Conclusion

Real-World Applications

Example Use Cases

Code Implementation

Conclusion

References

Further Reading

Frequently Asked Questions

Q: What is the binomial coefficient?

Q: How is the binomial coefficient calculated?

Q: What is the difference between the binomial coefficient and the factorial?

Q: Can you provide an example of how to use the binomial coefficient?

Q: How do you calculate the binomial coefficient for large values of n?

Q: Can you provide a Python implementation of the binomial coefficient?

Q: What are some real-world applications of the binomial coefficient?

Q: Can you provide some example use cases for the binomial coefficient?

Common Mistakes

Conclusion

Further Reading