What are the meanings of (2) (k≤n)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need to find 3 meanings.
![**Understanding Binomial Coefficients**
The expression \(\binom{n}{k}\) is read as "n choose k" and represents the binomial coefficient. It is used in combinatorics to denote the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection. The condition \(k \leq n\) ensures that the number of elements chosen does not exceed the total number of elements available.
### Formula for Binomial Coefficient:
The binomial coefficient is calculated using the formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
Where \(n!\) is the factorial of \(n\), which is the product of all positive integers up to \(n\).
### Applications:
1. **Probability**: In calculating the probabilities of different outcomes in binomial distributions.
2. **Algebra**: In expanding binomial expressions using the binomial theorem.
3. **Combinatorics**: In solving problems related to combinations and permutations.
By understanding the binomial coefficient, we can effectively analyze and solve a wide range of mathematical and real-world problems involving combinations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3d40ab3-26fa-4cc6-a20f-30a7f32ffb5d%2Fe171ae8a-df14-4d47-8c67-745902d35a55%2Fitw45vk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Binomial Coefficients**
The expression \(\binom{n}{k}\) is read as "n choose k" and represents the binomial coefficient. It is used in combinatorics to denote the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection. The condition \(k \leq n\) ensures that the number of elements chosen does not exceed the total number of elements available.
### Formula for Binomial Coefficient:
The binomial coefficient is calculated using the formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
Where \(n!\) is the factorial of \(n\), which is the product of all positive integers up to \(n\).
### Applications:
1. **Probability**: In calculating the probabilities of different outcomes in binomial distributions.
2. **Algebra**: In expanding binomial expressions using the binomial theorem.
3. **Combinatorics**: In solving problems related to combinations and permutations.
By understanding the binomial coefficient, we can effectively analyze and solve a wide range of mathematical and real-world problems involving combinations.
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