Final answer:
The number of different ways to choose a dozen donuts from the 4 varieties at a donut shop so that you get at least two donuts of every variety is 430467216.
Step-by-step explanation:
The question asks how many different ways there are to choose a dozen donuts from 4 varieties at a donut shop, with at least two donuts of every variety. To solve this problem, we can use the concept of combinations. We have 4 varieties of donuts, and we need to choose at least 2 donuts from each variety. This means we need to choose 2 donuts from each of the 4 varieties, so a total of 8 donuts. We then have 4 remaining donuts that we can choose from any of the varieties. So, the number of different ways to choose a dozen donuts is the product of the number of ways to choose 2 donuts from each of the 4 varieties (which is 4 choose 2 raised to the power of 4) and the number of ways to choose the remaining 4 donuts from any of the varieties (which is 4 raised to the power of 4). This gives us a total of (4 choose 2)^4 * 4^4 = 36^4 * 4^4 = 1679616 * 256 = 430467216 different ways to choose a dozen donuts.