Solution using multinomial theorem raised to a negative power The number of ways of selecting exactly $ 4 $ fruits out of $ 4 $ apples, $ 5 $ mangoes, $ 6 $ oranges is...
A) $ 10 $
B) $ 15 $
C) $ 20 $
D) $ 25 $
I did the solution writing all the possible ways, I am getting $ 15 $, which is correct. However, there is a way to solve this with a multinomial expansion of the negative power. Please explain. 
The solution is: We need to find the $x^4$ in the expansion of $(1+x+x^2+x^3+x^4)^3$. I want to understand this in a detailed manner. My question is, why $x^4$? Why $(1+x+x^2+x^3+x^4)^3$ and not $$(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6)?$$

Your suggestion of 
$$
(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6)
$$
is, technically, the correct expression. There are 5 mangoes and six oranges available, and this is reflected in $x^5$ and $x^6$ appearing.
However, the fact that there are $5$ and $6$ mangoes and oranges isn't really relevant, because we are only picking four fruits in total. So the problem would have the same answer if you just removed one mango and two oranges from the setup, and just had four of each fruit available.
We see this in the algebra too: the $x^5$ and $x^6$ terms in the above expression cannot possibly contribute to the final coefficient of the $x^4$ term. So they may be removed to simplify the expression.
