Calculating the Coefficient of x^34 in (1+x^2+x^7+x^16)^1000

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Discussion Overview

The discussion revolves around calculating the coefficient of \( x^{34} \) in the expression \( (1+x^2+x^7+x^{16})^{1000} \). Participants explore various methods and approaches to solve this combinatorial problem, referencing the multinomial theorem and providing examples.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks for the coefficient of \( x^{34} \) when \( n=1000 \) without providing prior work.
  • Another participant questions the original poster's understanding of the binomial and multinomial theorems, suggesting a lack of preparation for the problem.
  • A participant shares a method for calculating the coefficient by considering combinations of terms that sum to \( x^{34} \) and provides specific examples of combinations and their corresponding coefficients using factorials.
  • There is a mention of a simpler related problem involving \( x^3 \) in \( (1+x+x^2)^4 \) to illustrate the approach of finding coefficients.
  • Some participants express a need for additional resources or sample problems related to the binomial and multinomial theorems for study purposes.

Areas of Agreement / Disagreement

There is no consensus on the solution to the original problem, and multiple approaches are presented without resolution. Participants express varying levels of understanding and preparedness regarding the mathematical concepts involved.

Contextual Notes

Some participants may lack familiarity with the necessary mathematical theorems, which could affect their ability to engage with the problem effectively. The discussion includes various assumptions about the methods used to calculate coefficients.

FlyingMonkey
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(1+x^2+x^7+x^16)^n
What is the coeffient of x^34 when n=1000?

thanku
 
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what have you done so far?
 
gazzo said:
what have you done so far?
what ? so far ?
i stay at one place only. no farther, very near!
means i don no how to solve problem.
 
FlyingMonkey said:
what ? so far ?
i stay at one place only. no farther, very near!
means i don no how to solve problem.

Do you know anything about the "binomial theorem" or "multinomial theorem". If not, why in the world are you doing a problem like this?
 
Speaking of the multinomial theorem, does anyone know where i can get sample problems? I have a test next tuesday that is on the binomial and multinomial theorem, unfortunately my textbook has none.
 
Parth Dave said:
Speaking of the multinomial theorem, does anyone know where i can get sample problems? I have a test next tuesday that is on the binomial and multinomial theorem, unfortunately my textbook has none.
You may like to get your hands on Hall & Knight, if its possible for u to get it through some library maybe.
 
FlyingMonkey said:
(1+x^2+x^7+x^16)^n
What is the coeffient of x^34 when n=1000?

thanku

Well, to present a "hands on" start on this, you have to look at all the ways you can come up with X^34, and the exponents on these algebraic terms total 1000. (We need a lot of 1s.) One way is (X^16)^2(X^2)^1(1^997). (Hopefully you understand what I mean.) Coefficient for this is then (1000!)/[2!1!997!].

Then we proceed to look at (X^16)^1(X^7)^2(X^2)^2(1^997), this gives us the coefficient: (1000!)/[1!2!997!].

So proceeding in this method to look at all cases, we then add up all the coefficients, and presumedly, that is the correct answer.

A simpler question would be to find the coefficient for X^3 on (1+X+X^2)^4 .

There is only two ways: (X)^1(X^2)^1(1^2), and (x)^3(1^1). So the coefficient is 4!/(2!1!1!) and 4!/(3!1!) = 12 + 4 = 16.

Check out: http://encyclopedia.thefreedictionary.com/Trinomial theorem
 
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