Class 11 Exam > Class 11 Questions > Hey, I couldn't find the answer to this quest...
Start Learning for Free
Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question?
Most Upvoted Answer
Hey, I couldn't find the answer to this question "Summation of r^3 × n...
Community Answer
Hey, I couldn't find the answer to this question "Summation of r^3 × n...
Understanding the Summation
The summation of r^3 × nCr from 0 to n represents a combinatorial identity involving binomial coefficients. Here, r is a constant, and nCr (or C(n, r)) is the binomial coefficient representing the number of ways to choose r elements from a set of n elements.
Expression of the Summation
The expression can be written as:
- S = Σ (r^3 × nCr), where the summation runs from r = 0 to n.
Using Binomial Theorem
The binomial theorem states that:
- (x + y)^n = Σ (nCr × x^r × y^(n-r))
By differentiating this theorem, we can find relationships involving powers of r.
Applying the Identity
For the case of r^3, we can derive:
- S = n(n-1)(n-2)(1 + 1)^n
- This comes from applying differentiation three times to (1 + x)^n.
Final Result
The final result for the summation is:
- S = n^2(n + 1) / 4
This formula effectively summarizes the contribution of r^3 multiplied by the binomial coefficient over the range from 0 to n.
Conclusion
In summary, the summation of r^3 × nCr from 0 to n can be computed using binomial identities and differentiation techniques. The result provides a powerful tool for combinatorial problems and is rooted in the fundamental principles of binomial expansions.
The summation of r^3 × nCr from 0 to n represents a combinatorial identity involving binomial coefficients. Here, r is a constant, and nCr (or C(n, r)) is the binomial coefficient representing the number of ways to choose r elements from a set of n elements.
Expression of the Summation
The expression can be written as:
- S = Σ (r^3 × nCr), where the summation runs from r = 0 to n.
Using Binomial Theorem
The binomial theorem states that:
- (x + y)^n = Σ (nCr × x^r × y^(n-r))
By differentiating this theorem, we can find relationships involving powers of r.
Applying the Identity
For the case of r^3, we can derive:
- S = n(n-1)(n-2)(1 + 1)^n
- This comes from applying differentiation three times to (1 + x)^n.
Final Result
The final result for the summation is:
- S = n^2(n + 1) / 4
This formula effectively summarizes the contribution of r^3 multiplied by the binomial coefficient over the range from 0 to n.
Conclusion
In summary, the summation of r^3 × nCr from 0 to n can be computed using binomial identities and differentiation techniques. The result provides a powerful tool for combinatorial problems and is rooted in the fundamental principles of binomial expansions.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
Explore Courses for Class 11 exam
|
|
Similar Class 11 Doubts
Top Courses for Class 11View all
Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question?
Question Description
Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question?.
Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question?.
Solutions for Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? defined & explained in the simplest way possible. Besides giving the explanation of
Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question?, a detailed solution for Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? has been provided alongside types of Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? theory, EduRev gives you an
ample number of questions to practice Hey, I couldn't find the answer to this question "Summation of r^3 × ncr from 0 to n?" , Please help out by answering this question? tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup