Application of multinomial theorem in combinatorics I'm not understanding the method of using multinomial theorem in combinatorics problems.
For example, suppose we want to distribute $17$ identical oranges among $4$ children such that each child gets at least $1$ orange, how many ways can u distribute  the oranges?
The solution in the book says the max no of oranges a boy can get is $17$, so the required no of ways is 
Coefficient of $x^{20}$ in $[ x + x^2 + x^3 + \ldots + x^{17} ]^4$
If I am not right, this is a multinomial expansion, but what is the intuition behind this step? 
PS : I don't have a very high level of knowledge of math, so a simple intuitive explanation will be highly appreciated
 A: $x^{20}$ should be $x^{17}$ in your question.
$$(x+x^2+\cdots+x^{17})^{4}$$
$$=(x+x^2+\cdots+x^{17})\times (x+x^2+\cdots+x^{17})\times (x+x^2+\cdots+x^{17})\times (x+x^2+\cdots+x^{17})$$
So, you choose $x^A, x^B, x^C, x^D$ from each $()$ from the left to the right.
Then, what you need is the number of a set $(A,B,C,D)$ such that
$$A+B+C+D=17\ \text{and}\ A,B,C,D\ge 1$$
Note that the latter is already satisfied.
Each represents the number of oranges which a child get. Then, imagine when you find the coefficient of $x^{17}$. You'll choose every pattern of $(A,B,C,D)$ such that $A+B+C+D=17$.
Hence, the coefficient of $x^{17}$ represents the ways you can distribute the oranges. I hope this is helpful.
A: First off: if a kid gets 17 oranges then the other kids die of hunger.And that one kid gets too much energy and becomes annoying.
Here is how I would approach the problem. First give 1 orange to each kid. You now have 13 oranges which you can distribute however you like.
Now use stars and bars to see in how many ways you can distribute the remaining 13 oranges.(see this answer)
Here there are 3 bars and 13 stars. So there are $\binom{16}{3}$ ways to distribute the oranges.
Hope this helps.
