Correct.
I need assistance in writing/answering using the proper text for Discrete Math/Statistical Math. Here is the problem:
A building supplies store carries metal, wood, and plastic moldings. Metal and wood moldings comes in two different colors. Plastic moldings comes in six different colors.
a.) How many choices of molding does this store offer?
On the first glance, I would have answered 3; however, I am assuming they are counting each color as well. I know metal=2, wood=2, and plastic=6. Making this 2+2+6=10
b.) If each kind and each color of molding comes in four different lengths, how many choices does the consumer have in the purchase of one piece of molding?
Metal=1(color)x4(lengths)+1(color)x4(lengths)=8 choices (long version)
Wood=2(color)x4(lengths)=8 choices
Plastic=6(color)x4(lengths)=24 choices
8+8+24=40 total choices
Well, this totally depends on the requirements of the course. There is no one way to write this more formally. In fact, it would not help readability if the proof in the OP is written in symbols.
But you could declare, for example, the following notations. Let M = {m_{1}, m_{2}}, W = {w_{1}, w_{2}} and P = {p_{1}, ..., p_{6}} where m_{i}, w_{i} and p_{i} are the metal, wood and plastic moldings, respectively, of color i. Obviously, |M| = |W| = 2 and |W| = 6. Also, these sets are pairwise disjoint. Question a) asks to find |M ∪ W ∪ P| = |M| + |W| + |P|. Question b) asks to find |(M ∪ W ∪ P) x {1, 2, 3, 4}| where the second factor is the set of lengths.