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
Best Answer
$x^{20}$ 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.