Combinations of pizza toppings with at least one vegetable and at least one meat. Here is a question from my quiz:

Superior Pizza has seven vegetable ingredients and nine meat ingredients. 
  The number of ways to select five ingredients (no doubling on ingredients) 
  with at least one vegetable ingredient on a pizza is:

And there are multiple choice solutions:


*

*$4242$

*$4221$

*$4368$

*$4347$

*None of the above.


Similarly, how many ways can we select five ingredients with at least one meat and one vegetable?
The way I tried to approach the first one (and second, as they are similar) is to take the following:
$$ {7 \choose 1} \cdot P(15 \space\space 4) $$
I did this, because I know I need 1 vegetable, and the rest can be any of the remaining $15$ ingredients.
However, I got an absurdly large answer that I don't think is correct. I don't have the correct answer yet, but I know that I got an $8/10$ on the quiz and I believe these two are the ones I did wrong. Can anyone help me solve this properly?

For the first question, you want combinations, not permutations, as you only care what toppings are on the pizza.  If you select one vegetable first, you will double count the pizzas with two vegetables, once where each vegetable is the first one. A better approach is to count the total number of five item pizzas, then subtract the ones that have no vegetables.  
