Multinomial Expansion

1. Jun 21, 2011

Mentallic

1. The problem statement, all variables and given/known data
1) Find the coefficient of $$w^2x^5y^7z^9$$ in $$(w+x+y+z)^{23}$$
2) When $$(w+x+y+z)^{23}$$ is expanded and terms collected, how many different terms will there be?

3. The attempt at a solution
Again, I'm studying through my text book and I'm finding these kinds of questions in past exams that aren't present in the book. I'm unsure how to begin.

2. Jun 21, 2011

micromass

Staff Emeritus
Hi mentallic!

See http://en.wikipedia.org/wiki/Multinomial_theorem

Basically, it states that

$$(x_1+...+x_m)=\sum_{k_1+...+k_m=n}{\binom{n}{k_1,...,k_m}x_1^{k_1}...x_m^{k_m}}$$

So what does that give us in our specific case? Here: m=4 and n=23.

3. Jun 21, 2011

Mentallic

Oh thanks, that makes finding the answer very simple!

But the multinomial expansion isn't in our syllabus so I'm guessing we need to argue with separate combinatoric multiplications.

Judging by the multinomial expansion though, I'm guessing the second last step in the solution would be of the form:

$$\binom{23}{2}\binom{23-2}{5}\binom{23-2-5}{7}\binom{23-2-5-7}{9}=\frac{23!}{2!5!7!9!}$$

Somehow I need to be able to argue why the coefficient is equal to these combinations?

And I still have no clue how to answer part 2.

4. Jun 21, 2011

micromass

Staff Emeritus
OK, the things that you could do then is actually show the multinomial theorem in the case m=4. This is very easy and natural because it just requires you to use the binomial theorem a few times.

$$(y+z)^{23}$$

this is just the binomial theorem. Then we add one more term:

$$(x+(y+z))^{23}$$

this is again using the binomial theorem and previous result. And finally, we have

$$(w+(x+y+z))^{23}$$

Working this out should give you the statement of the multinomial theorem.

Now, this statement is also needed in part 2, since that basically asks you to calculate in how many ways we can write $k_1+k_2+k_3+k_4=n$.

5. Jun 21, 2011

Mentallic

Hmm, since we haven't diverged from the multinomial theorem, I'll use it, but I don't know how happy they'll be about me using a different technique to the one they taught in the class that I missed.

Oh really? Ok well that I can answer, but I don't quite understand why exactly. I'll have to come back to this detail after my exam is over.

Thanks a lot for your help micromass!

6. Dec 2, 2011

NguyenSP

When (w+x+y+z)23 is expanded and terms collected, how many different terms will there be?

from pascal's triangle you can find $\frac{26!}{23!3!}$= 5200 diffenent terms.

Last edited: Dec 2, 2011