29. The given sum is equal to (m₂ + my + ms). 2+m³). n To see this, compute the coefficient of " in each side of (1+x)(1+x)(1+x)m³ = (1+x)m₁+m₂+ms In this computation use the binomial theorem.
29. The given sum is equal to (m₂ + my + ms). 2+m³). n To see this, compute the coefficient of " in each side of (1+x)(1+x)(1+x)m³ = (1+x)m₁+m₂+ms In this computation use the binomial theorem.
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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Question
The one with 29 written is the hint on how to solve
![**Problem Statement:**
*Find and prove a formula for:*
\[
\sum_{r, s, t \geq 0, \, r + s + t = n} \binom{m_1}{r} \binom{m_2}{s} \binom{m_3}{t}
\]
*where the summation extends over all nonnegative integers \( r, s, \) and \( t \) with sum \( r + s + t = n.***](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe21428a7-a234-45c9-b550-85f6c9ea7218%2F731af56d-3be1-4188-b8ce-0144b6bbc191%2Fdyxd5i_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
*Find and prove a formula for:*
\[
\sum_{r, s, t \geq 0, \, r + s + t = n} \binom{m_1}{r} \binom{m_2}{s} \binom{m_3}{t}
\]
*where the summation extends over all nonnegative integers \( r, s, \) and \( t \) with sum \( r + s + t = n.***
![29. The given sum is equal to
\[
\binom{m_2 + m_3}{n}.
\]
To see this, compute the coefficient of \(x^n\) in each side of
\[
(1 + x)^{m_1} (1 + x)^{m_2} (1 + x)^{m_3} = (1 + x)^{m_1 + m_2 + m_3}.
\]
In this computation use the binomial theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe21428a7-a234-45c9-b550-85f6c9ea7218%2F731af56d-3be1-4188-b8ce-0144b6bbc191%2Fvbq086_processed.png&w=3840&q=75)
Transcribed Image Text:29. The given sum is equal to
\[
\binom{m_2 + m_3}{n}.
\]
To see this, compute the coefficient of \(x^n\) in each side of
\[
(1 + x)^{m_1} (1 + x)^{m_2} (1 + x)^{m_3} = (1 + x)^{m_1 + m_2 + m_3}.
\]
In this computation use the binomial theorem.
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