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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?
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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.
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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?.
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