How many flags, with 3 horizontal stripes, can be made if two stripes are one color and the third is a different color? For example, take the Austrian flag (red, white, red). I thought that the answer would be three from $\frac{3!}{1!2!}$, but the instructor said twelve, because from the product rule: $3\cdot2\cdot2$. Can anyone give me an explanation? 
Thanks.
 A: If there are only 2 colours to choose from, then there are only 6 ways. There are $\binom{3}{1}=3$ ways to choose which stripe will be the single one. You then have two choice for the colour of that single stripe, and there is then no choice for the colour of the other two stripes as they must be the other colour. This gives $\binom{3}{1}*2=6$ flags.
Another way of counting it is: There are two choices of colour for each stripe, for a total of $2^3=8$ possible flags. However, this includes the $2$ flags where all the stripes are the same colour. That leaves $2^3-2=6$ flags which use both colours (2 stripes of one colour, 1 of the other).
A: Talking about flags: I think it is safe to assume that, for example inverting the top and bottom red stripes in the Austrian flag, does not give a new flag.
Having said this, I struggle to follow your instructor's solution, and also the answer posted above by the fellow with original views on last century's history (now edited, and confirming my views, on flags that is, not history...).
I cannot see nothing but six possible flags, by selecting where to place the colour available only on one stripe, and which, out of the two available, such colour is, so:
1) W R R
2) R W R
3) R R W
and additional three by inverting red and white.
A: Of the three stripes two which are of the same color can be chosen in $^3C_2$ way, the can be filled with color in $2$ ways as there are two colors and the last different color stripe can be filled in $2$ ways. Total being $3 \times2 \times 2=12$.
However this will repeat situation when all flag is the same color 3 times each we drop the repeating ones and it is 8 in total. If Red(R) and white(W) are the colors all possible flags are. 
RRR, WWW, WWR, RRW, RWR, WRW, WRR, RWW
