Combinatorics Reasoning So i was given a question that begins like this.

How many ways are there to buy $10$ pieces of candy from an (unlimited)
  supply of three kinds: jelly beans, chocolate almonds and skittles with at least $2$ candies of each kind.

I'm new to combinatorics so i'm really lost on how to solve this.
 A: 
How many ways are there to buy 10 pieces of candy from an (unlimited) supply of three kinds: jelly beans, chocolate almonds and skittles with at least 2 candies of each kind.

First buy two of each kind, then determine how many ways to buy the remaining four.


*

*Four of one kind: 3 ways: 

*Three of one kind and one of another: $\binom 3{1}\binom 21$ ways

*Two of one kind and two of another: $\binom 32$ ways

*Two of one kind and one of each other: $\binom{3}{1}\binom 22$ ways


$$3+6+3+3=15$$

Using "stars and bars" we count ways to arrange four 'stars' and two 'bars' (to count ways to select four candies from three 'boxes'): $\binom {4+2}{4}=15$
A: 
Algebraic answer but one should be able to visualize

Call the chocolate types J, C and S. We will assume the chocolates get delivered in a bag without any inherent order.
Take a look at this (apparently unrelated) expression: 
${
(j^2+j^3+j^4+\ ...)
(c^2+c^3+c^4+\ ...)
(s^2+s^3+s^4+\ ...)
}$
assume that all the 3 vars are ${0<j,c,s<1}$
In the product, all the combinations of ${j,c,s}$ 
will appear which are of interest to us. They are those which look like these:

${j^p.c^q.s^r}$ 
such that 
${2 \leq p,q,r\leq 6}$ and ${p+q+r=10}$
${p,q,r}$ cannot be more than 6, otherwise their sum cannot be 10.
The number of ways can be computed by substituting ${x}$ for each of ${j,c,s}$ in the above expression and calculating the coefficient of ${x^{10}}$.
The fact that these are infinite series do not matter, since any ${x^p}$ with ${p>6}$ will not contribute to ${x^{10}}$.
The expression becomes:
${(x^2+x^3+x^4+\ ...)^3\\
=x^6.(1+x+x^2+x^3+\ ...)^3\\
=x^6.(1-x)^{-3}\\
=x^6.(1+3x+\frac{3.4}{1.2}x^2+\frac{3.4.5}{1.2.3}x^3+\frac{3.4.5.6}{1.2.3.4}x^4+\ ...)\\
}$
The coefficient of ${x^{10}}$ is 
${\frac{3.4.5.6}{1.2.3.4}=15}$
