Making of a committee and other combination questions.

[kakariki]

Hey!
I am having trouble doing this question. I have gone through my notes and there is nothing entirely similar to this problem, and I was hoping you people could help me out!
Question
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?
b) In how many ways could the committee choose up to 3 toppings for a pizza?
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.
d) For one of these cases, determine the number of ways of choosing and distributing the 10 toppings.

Solution
Whoo, that's a long question! I will try to explain what I have tried/done so far in each part of the question, I am not sure of any of these solutions, so I hope you can help me out!

a) Since it is a committee of 3, and it needs to have at least 1 vegetarian and 1 non vegetarian. I believe I can go: (7C1)(13C1)(18C1). My logic behind this is: (7C1) will give one vegetarian on the committee; (13C1) will give one non vegetarian on the committee; (18C1) is 20 minus the two that are already on the committee, and I need one more member from that group. Does this make sense? I get the solution of 1638 different committee combinations.

b) I understand this to mean "how many 3 topping combinations are there from the 10 pizzas". So I get the solution of (10C3).

c) I do not really understand what part c is asking. I think it's asking what the combination of toppings is if you only order them each once, and you can have a max of 3 toppings on each pizza. Now, how would I go about solving this? Would likely be a fairly high number in that each topping could end up alone, with 1 other and with 2 others. This one I definitely need the most help with.

d) This relates to part c, so I think if I understand how to do c this will become clear.

Thanks for reading my thread, hopefully you will be able to help me out! I am new here (this is my first post) so please let me know if there is a better way of asking for help!

Thanks!

 

[soroban]

Hello, kakariki!

I don't understand questions (c) and (d) . . .

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.
The pizza shop 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?

\(\displaystyle \text{There are: }\:\text{7 V's (vegetarians) and 13 C's (carnivores).}\)


\(\displaystyle \text{There are two possible committee compositions:}\)

. . \(\displaystyle \text{1V, 2C: } \;(_7C_1)(_{13}C_2) \;=\;7\cdot78 \:=\:546\)

. . \(\displaystyle \text{2V, 1C: } \;(_7&C_2)(_{13}C_1) \;=\;21\cdot13 \;=\;273\)

\(\displaystyle \text{Therefore, there are: }\:546 + 273 \:=\:819\text{ possible committees.}\)

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

. . \(\displaystyle \begin{array}{ccccccc}\text{0 toppings:} & _{10}C_0 &=& 1 \\ \text{1 topping:} & _{10}C_1 & =& 10 \\ \text{2 toppings:} & _{10}C_2 &=& 45 \\ \text{3 topping:} & _{10}C_3 &=& 120 \end{array}\)

\(\displaystyle \text{There are: }\;1 + 10 + 45 + 120 \;=\;176\text{ possible choices of toppings.}\)

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

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