We have three responses for you
When you ask such a questions about your work, it is good to indicate your steps. This makes it easier to see where you went wrong, if you did, but also to see the bigger part where you got it right.
You did not indicate any of your work, so I started by taking your answer and factoring it:
3276 = 2*2*3*3*7*13
It is easy to see where you got the 7: it is the number of possible choices for Rice/Noodles. Also, 2*3 = 6 is the number of possible choices of Sauce. So I suppose your answer is of the form
7 * (something) * 6.
Up to now, you are correctly using the multiplication principle, and you have two of the three terms correct. Not bad! However as you feared the number of choices of Main Ingredient is incorrect. You have there
6*13 = 12*13/2 = 1 + 2 + 3 + ... + 11 + 12.
In this case it is the wrong formula. If there were two main ingredients, say Asparagus (A) and Beets (B) there would be 3 choices: A, B or AB. If there were a third ingredient, say Carrots (C), there would be seven choices: A, B, C, AB, AC, BC, ABC. What is the number of choices corresponding to twelve ingredients?
If I had to grade this in an assignment question, I'd give 0 for a wrong answer like 3276. However if all the steps were there to explain the answer of 7*78*6 = 3276, neatly written with proper grammar, I'd give part marks, probably four to seven out of ten. So it is important to indicate your steps. Unfortunately not enough students do this, and teachers end up having to spend too much time arguing with students who got 0 and find it unfair, rather than dealing with students who want to understand and learn.
Finally, multiply the number of choices for rice/noodles (7) by the number of choices for sauces (6) by the number of choices for mains (2^12 - 1) and you have the total number of combinations.
Yes, definitely too small.
First, how many choices for main dishes are there? Try a simpler version with only 2 main dishes and try to list all possibilities. See a pattern? Now try the pattern with 3 to see if it works and if so, apply it to 12. [The answer to this should be roughly 800 larger than the answer you got.]
Now, you have seven choices for rice/noodles with each of these collections of mains. And with each of those meals, six choices for sauce.
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