Counting using products.

Today I proved the binomial theorem for my class. I did an informal proof showing them how, if we consider the set containing the alphabet: S = \{ a,b ,c, d, ..., z \} and the following product.

\displaystyle \prod_{s \in S} (1+s_i) = (1+a)(1+b)(1+c)\cdots (1+z)

expanding this produces a sum with every combination of the letter a-z:

\displaystyle \prod_{s \in S} (1+s_i) = 1 + a + b + \cdots + z + ab + ac + ad + \cdots + az + bd + be + \cdots + bz + \cdots + abc + abd + \cdots + abcd + \cdots + abcdefghijklmnopqrstuvwxyz

if we count all of the terms in this sum with, say three elements, we have the number combinations of 26 items taken 3 at a time or 26 \choose 3. Next, we make every letter x:

(1+x)^{26} = 1 + x + x + \cdots + x + xx + xx + xx + \cdots + xx + xx + xx + \cdots + xx + \cdots + xxx + xxx + \cdots + xxxx + \cdots + xxxxxxxxxxxxxxxxxxxxxxxxx

(Pretty silly looking I know!) But, now we can see that the coefficients of the terms in the expansion are given by the same formula used to count combinations in probability.

That was today's lesson. What I've noticed is that this manner of counting is the same basic idea as what I've been looking at with integer partitions. (There we use a different product and coefficents to count.)

I wonder if this type of process has a name, or if it is further formalized in any contexts?