Coefficient

[sayo]

What is the coefficient of the x^111 term in the expansion?

 

[JeffM]

You should know by now that you get help after you show work.

Do you know the binomial theorem?

 

[sayo]

You should know by now that you get help after you show work.

Do you know the binomial theorem?

i am sorry. of course i know binomial theorem. we are learning that in 10th grade but i coudn't do anything

 

[Dr.Peterson]

i am sorry. of course i know binomial theorem. we are learning that in 10th grade but i coudn't do anything

In order to help, we need to see what you know, and where you are stuck.

Of course, this is not a mere binomial, so there is something more going on beyond the binomial theorem. It will be really helpful if you can show us the theorem(s) you have learned, so we can see what you can use.

But here is a separate idea you may need to turn to, which perhaps relates less to a theorem you learned than to the ideas that went into proving them: Imagine writing out the whole polynomial and expanding it. What would it take to get a term with x^111?

Or, try actually doing that with a smaller example. In the expansion of (1 + x + x^2 + x^3)^3, what is the coefficient of x^4? Think about how you could find that number without actually writing it out; what terms combine to give it? Think in terms of combinatorics.

 

[JeffM]

OK I shall give you a hint

[MATH]x \ne 1 \implies (1 + x + x^2 + ... x^{98} + x^{99} + x^{100}) = \dfrac{1 - x^{101}}{1 - x}[/MATH]
If you not see where this comes from, think about this:

[MATH]\dfrac{1 - x^{101}}{1 - x} = \dfrac{1 - (x - x) - (x^2 - x^2) ... - (x^{100} -x^{100}) - x^{101}}{1 - x} =[/MATH]
[MATH]\dfrac{(1 - x) + (x - x^2) + (x^2 - x^3) ... + (x^{100} - x^{101}}{1 - x} =[/MATH]
[MATH]\dfrac{1(1 - x) + x(1 - x) + x^2(1 - x) ... + x^{100}(1 - x)}{1 - x} =[/MATH]
[MATH]\dfrac{(1 - x)(1 + x + x^2 ... + x^{100})}{1 - x} =[/MATH]
[MATH]\dfrac{\cancel{(1 - x)}(1 + x + x^2 ... + x^{100})}{\cancel{1 - x}} =[/MATH]
[MATH]1 + x + x^2 ... + x^{100}.[/MATH]

 

[pka]

i am sorry. of course i know binomial theorem. we are learning that in 10th grade but i coudn't do anything

To Sayo, please tell us about the topics that you have studied in connection with this assignment.
I cannot tell what language it posted problem is written. You said it is tenth grade level.
So what methods are you expected to bring to this assignment?
If it were the one Prof. Peterson posted or if it were to find the coefficient of \(x^8\) in \(\left(1+x+x^2+x^3+x^4\right)^{3}\)
I would agree those tasks are reasonable.
BUT to ask a tenth grade class to find \(Kx^{111}\) in the expansion \({\left( {\sum\limits_{k = 0}^{100} {{x^k}} } \right)^3}\) is unreasonable.
Please tell us more about your class.

 

[Steven G]

You are multiplying three factors. In fact you are multiplying every combination where you take one term from each of the three factors and multiply them out.
Now you have to add the exponents of each term you pick from each factor based on what the three factors look like.
Recall that 1 = 1x⁰.
You need to count the number of ways that you can choose three numbers (repeats are allowed-why?)from the set 0, 1, 2, ..., 100 so the three numbers add up to 101.

 

[lex]

There are several ways to do this.
It is an example of 'stars and bars' (if you are familiar with this).
[MATH](1+x+ .... + x^{100})(1+x+ .... + x^{100})(1+x+ .... + x^{100})[/MATH]We want to pick a term from each bracket so that when we multiply the three terms together, the powers add up to 111.
We want to count all such combinations.
I.e. how many integer solutions are there to:
[MATH]x_1+x_2+x_3=111[/MATH]with [MATH]0≤x_i≤100[/MATH]
Assuming you are familiar with 'stars and bars', here is a similar (but more difficult) example.




Note: your example is much simpler, since there can be a maximum of one of the variables >100.

 

[Steven G]

There are several ways to do this.
It is an example of 'stars and bars' (if you are familiar with this).
[MATH](1+x+ .... + x^{100})(1+x+ .... + x^{100})(1+x+ .... + x^{100})[/MATH]We want to pick a term from each bracket so that when we multiply the three terms together, the powers add up to 111.
We want to count all such combinations.
I.e. how many integer solutions are there to:
[MATH]x_1+x_2+x_3=111[/MATH]with [MATH]0≤x_i≤100[/MATH]
Assuming you are familiar with 'stars and bars', here is a similar (but more difficult) example.

View attachment 28062


Note: your example is much simpler, since there can be a maximum of one of the variables >100.

That is what I wrote!

 

[lex]

That is what I wrote!

Sorry - I didn't intend to steal your idea just to add to it by making explicit that this is an example for 'stars and bars' and further, giving an example where there is a maximum to the value each variable can take.
If attempted this way, this example requires 'stars and bars' and PIE.

You need to count the number of ways that you can choose three numbers (repeats are allowed-why?)from the set 0, 1, 2, ..., 100 so the three numbers add up to 101.

(Also they add up to 111, not 101)!

 

[Steven G]

Sorry - I didn't intend to steal your idea just to add to it by making explicit that this is an example for 'stars and bars' and further, giving an example where there is a maximum to the value each variable can take.
If attempted this way, this example requires 'stars and bars' and PIE.


(Also they add up to 111, not 101)!

Just having some fun at your expense.

 

[lex]

Good! I didn't tread on your beard then!

 

[Steven G]

Good! I didn't tread on your beard then!

Not at all!