Multinomail Theorem on Wikipedia

1. Aug 13, 2011

Saitama

1. The problem statement, all variables and given/known data
Hi!! This is not a homework, i am just trying to understand the multinomial theorem on Wikipedia.
http://en.wikipedia.org/wiki/Multinomial_theorem

I don't understand how to apply this formula:-

I am trying to apply this formula in this question:-
(1+x+x2)5
and i am trying to solve it by the multinomial theorem.

Firstly i don't understand what are these k1,k2, k3.....

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: Apr 26, 2017
2. Aug 13, 2011

I like Serena

Your k1,k2, k3, ... are any combination of numbers that add up to n.

The capital sigma is the regular summation symbol meaning that should sum the terms for any combination of ki's that you can think of that add up to n.

The capital pi is the product symbol, meaning that you should take the product of the factor for any t that is between 1 and m.

3. Aug 13, 2011

I like Serena

As for constructing a triangle.
You can construct Pascal's triangle by building up:
1
(1+x)
(1+x)2 = 1 + 2x + x2
(1+x)3 = 1 + 3x + 3x2 + x3
(1+x)4 = ...

You can do the same for (1+x+x2)n.
The coefficients you will find are the multinomials.

Btw, my time is up. Have to run now!

4. Aug 13, 2011

Saitama

Could you please provide an example?

Yep!! I can do that but that would be more time consuming. So i thought that i should learn this theorem. And also day after tomorrow, its my maths exam, so i thought it would be very useful for that.

(Ok, bye )

5. Aug 13, 2011

Icetray

I'm very keen to learn this as well. An example would be very much appreciated. (:

6. Aug 13, 2011

Saitama

Expand (x+y)n.
Then substitute x=1 and y=x.
The equation which you would get is like this:-
$$(1+x)^n={}^nC_0+{}^nC_1x+{}^nC_2x^2+.........+{}^nC_nx^n$$

We have to solve this equation:- (1+x+x2)n
For that we need to substitute x=1 and y=(x+x2) in (x+y)n expansion.

But that would be very time consuming.

7. Aug 13, 2011

I like Serena

Well, here's a general example for (A+B+C)n:
http://en.wikipedia.org/wiki/Pascal's_pyramid#Trinomial_Expansion_connection
It shows all the multinomials for this one expression in a triangle.
But I'm afraid it's still pretty messy.

Here's another example:
http://en.wikipedia.org/wiki/Multinomial_theorem#Example_multinomial_coefficients
However, I do not think I can explain it simpler.

As for the (1+x+x2)n.
There is a simple pattern.
It's simpler because it does not contain all the multinomial coefficients, but summations of them.
To find it, I think it's easiest to work out the first 3 layers of the pyramid.
You should be able to see the pattern then.

8. Aug 13, 2011

Saitama

I just want to understand this formula:-

If i try to solve this question (1+x+x2)5 using the multinomial theorem, should i go like this:-
$$(1+x+x^2)^5=\sum_{0+1+2=3}\frac{5!}{0!1!2!}\Pi_{1 \leq t \leq m} x^{k_t}_t$$

Now what should i do next?

9. Aug 13, 2011

SammyS

Staff Emeritus
You need k1 + k2 + k3 = 5, NOT 3. In other words, once you pick k1 & k2, then you must have k3 = 5 - (k1 + k2) .

You can think of the sum as having the following:
k1 = 0 to n
k2 = 0 to n - k1
k3 = 0 to n - (k1 + k2)

10. Aug 13, 2011

Saitama

Hi SammyS!!

Ok, i am trying it out again:-

$$(1+x+x^2)^5=\sum_{2+1+2=5}\frac{5!}{2!1!2!}\Pi_{1 \leq t \leq m} x^{k_t}_t$$
Is it ok now?

11. Aug 14, 2011

I like Serena

It should be:
$$(1+x+x^2)^5=... + \frac{5!}{2!1!2!}(1^2 x^1 (x^2)^2) + ...$$

The total number of terms is 21.

12. Aug 14, 2011

Saitama

Where's the sigma?

13. Aug 14, 2011

I like Serena

The sigma is a shorthand notation to indicate a summation.
When written out in terms, there is no sigma any more.
Of course I did not write out all of the terms - too much work!

14. Aug 14, 2011

Saitama

I am still not able to get what happens to $$\sum_{2+1+2=5}$$

15. Aug 14, 2011

I like Serena

Let me give a simpler example.
$$\sum_{k+m=2} k \cdot m = 0 \cdot 2 + 1 \cdot 1 + 2 \cdot 0$$

Does that help?

16. Aug 14, 2011

Saitama

Yes that helped.
But then what we have to do with this:-
$$\Pi_{1 \leq t \leq m} x^{k_t}_t$$
I don't know what's this symbol called and i have never dealt with it. I only know what is its function.
If it is given like this:-
$$\Pi_{i=1}^{n}a_i=a_1 \cdot a_2 \cdot a_3........... \cdot a_n$$

17. Aug 14, 2011

I like Serena

The symbol is a capital pi. It is the product symbol.

As an example:
$$\prod_{1 \leq t \leq 3} x^{k_t}_t = x_1^{k_1} \cdot x_2^{k_2} \cdot x_3^{k_3}$$

18. Aug 14, 2011

Saitama

Thanks fro helping me to understand the Multinomial theorem but that is still a clumsy method to solve this question:-
$$(1+x+x^2)^5$$
Isn't there any easy method?

19. Aug 14, 2011

I like Serena

In a previous post I wrote:

I haven't worked out a formula yet, but I can see a simple pattern, similar to the regular Pascal's triangle.
Based on this pattern I can probably work out a formula, but so could you!