Help to understand and apply the multinomial theorem

Question

I am reading about the multinomial theorem here

How do I read the summation notation in this line:

Also, can someone please show me how to apply it to the following expansion:

$$\left( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}\right)^4$$

I am not sure how to map the notation into the actual expression.

Answer 1

The multinomial notation means that this is the sum over all possible values $i_1,i_2,\dots, i_n$ for which $i_1+i_2+\dots +i_n=N$ holds. This means that, for $n=2$ and $N=3$, you have the values $0,3$, $1,2$ $2,1$ and $3,0$, meaning that the sum in this case would contain the sumands $$\frac{3!}{3!0!}x_1^3x_2^0+\frac{3!}{2!1!}x_1^2x_2^1 + \frac{3!}{1!2!}x_1^1x_2^2+\frac{3!}{0!3!}x_1^0x_2^3. $$

In your case, you have $N=4$ and $n=4$, so you have some more combinations, namely $34$. This gives you $34$ different values to sum up. It's a mess, but doable.

The $34$ combinations are: $(4,0,0,0)$, $(0,4,0,0)$, $(0,0,4,0)$ and $(0,0,0,4)$ to begin with, then all possibilities of $4=3+1$, of which there are $12$:

$3,1,0,0$

$3,0,0,0$

$3,0,1,0$

$3,0,0,1$

$1,3,0,0$

$0,3,1,0$

$0,3,0,1$

$1,0,3,0$

$0,1,3,0$

$0,0,3,1$

$1,0,0,3$

$0,1,0,3$

$0,0,1,3$

then all possibilities for $4=2+1+1$, again there are $12$ of them:

$2,1,1,0$

$2,1,0,1$

$2,0,1,1$

$1,2,1,0$

$1,2,0,1$

$0,2,1,1$

$1,1,2,0$

$1,0,2,1$

$0,1,2,1$

$1,1,0,2$

$1,0,1,2$

$0,1,1,2$

and another $6$ possibilities for $4=2+2$:

$2,2,0,0$

$2,0,2,0$

$2,0,0,2$

$0,2,2,0$

$0,2,0,2$

$0,0,2,2$

Edit: One last possible way of writing $4$ as a sum of $4$ integers is of course $4=1+1+1+1$, yielding a total of $35$ possible ways.

Answer 2

There is no need to remember the formula. Its a consequence of simple combinatorics. I assume you know that if you are given $n_1$ objects of type $1$, $n_2$ objects of type $2$,... and $n_k$ objects of type $k$ with $n_1 + \dots n_k = N$, then there are $\frac{N !}{n_1 ! \dots n_k !}$ distinct arrangements.

The same happens when you consider $(x_1 + \dots x_k)^N$. Every term in its expansion is of degree $N$ and if you need to find the coefficient of say $x_1 ^{i_1} \dots x_k ^{i_k}$, the problem is equivalent to the above problem. Using the same argument we can write the given formula.

So, given $N$, find different partitions of $N$ and for each partition write the given formula.