The student council is ordering pizza for their next meeting. There are 20 council members, 7 of whom are vegetarian. A committee of 3 will order 6 pizzas from a pizza shop that has a special price for large pizzas with up to three toppings. The shop offers 10 different toppings.

a) How many different pizza committees can the council choose if there must be at least 1 vegetarian and 1 non-vegetarian on the committee?

= C(13,2) C(7,1)
= (78)(7)

= C(13,1) C(7,2)
= (13)(21)
= 273

= 546+273
= 819

therefore 819 committees

b) In how many ways could the committee choose up to 3 toppings for a pizza?

= C(10,1) + C(10,2) + C(10,3)
= 10+45+120
= 175

c) The committee wants as much variety as possible in the toppings. They decide to order each topping exactly once and to have at least 1 topping on each pizza. Describe the different cases possible when distributing the toppings in this way.

Case 1:

2 pizzas with 1 topping
4 pizzas with 2 toppings

Case 2:

3 pizzas with 1 topping
2 pizzas with 2 toppings
1 pizza with 3 toppings

Case 3:

4 pizzas with 1 topping
2 pizzas with 3 toppings

d) For one of these cases, determine the number of ways of choosing and distributing the 10 toppings.

I'm having a difficult time understanding this question. I was hoping someone would be able to possibly help explain or steer me in the right direction?