A bagel shop has plain muffins, cherry muffins, chocolate muffins, almond muffins, apple muffins, and broccoli muffins. How many ways are there to choose:
1.) two dozen muffins with at least two of each kind?
2.) two dozen muffins with at least five chocolate muffins and at least three almond muffins?
3.) two dozen muffins with no more than two broccoli muffins?
4.) two dozen muffins with at least one plain muffin, at least two cherry muffins, at least three chocolate muffins, at least one almond muffin, at least two apple muffins and no more than three broccoli muffins?
Answers???
1.) $\left(\!\!{6\choose 12}\!\!\right)$ Since when order doesn't matter there is only one way to choose 12 muffins (2 for each flavor), there are 12 left to choose from out of the 6 flavors.
2.) $\left(\!\!{6\choose 16}\!\!\right)$ Same reasoning as #1.
3.) $\left(\!\!{5\choose 22}\!\!\right)$ We are choosing 2 broccoli and thus eliminating 1 flavor and then have only 22 choices left.
4.) $\left(\!\!{5\choose 12}\!\!\right)$ Same reasoning as the above but combined. There is only 1 way to choose at least one plain muffin, at least two cherry muffins, at least three chocolate muffins, at least one almond muffin, at least two apple muffins.  So we have 15 choices left, but then we are choosing at most 3 broccoli and so we have 5 flavors left with 12 choices.
Do these sound reasonable?

Your first two are fine. For the third one you’ve solved the wrong problem: you’ve calculated the number of ways of choosing two dozen muffins including exactly two broccoli muffins, when the problem asked for the number of ways of choosing two dozen muffins with at most two broccoli muffins. This simply adds another two terms to your answer:
$$\left(\!\!{5\choose 22}\!\!\right)+\left(\!\!{5\choose 23}\!\!\right)+\left(\!\!{5\choose 24}\!\!\right)$$
You’ve made a similar error in the fourth problem: you’ve counted the number of ways that include exactly $3$ broccoli muffins. You also need the ways that include $2,1$, or $0$ broccoli muffins.
(Broccoli muffins?)
