Permutation and combination question?

Question

The student council is having pizza at thier next meeting. There are twenty council members, eight of whom are vegetarians. A pizza committee of three will order from a pizza shop that has a special price for large pizzas. The shop offers nine different toppings. They want at least one topping on each pizza.

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

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

Please show work


Thanks

Answer 1

a) combination question. Three slots, first veg, second non-veg, third open. 8*12*18 / 3! = 8*12*18/6 = 288.

b) combination question (order of toppings doesn't matter presumably)
1 topping: 9 possibilities
2 toppings: 9! / 2!*7! = 9*8/2 = 36 possibilities
3 toppings: 9!/ 3!*6! = 9*8*7/6 = 84 possibilites

Total = 9+36+84 = 129.

Answer 2

If we treat all the vegetarians as identically-functioning people (not caring about which ones we get), and non-vegetarians as identically-functioning people, there are only two ways to select the committee:

veg, veg, non-veg
or
veg, non-veg, non-veg

However, if we treat the people as individuals, we have the same two choices of committee members, but now we assume that two different vegetarians represent different outcomes.

For the veg, veg, non-veg committee, there are

(8-choose-2) * (12-choose-1) ways to choose. Since choosing vegetarian A and B is the same as choosing B and A (we don't care about the order in which they're chosen), we use combinations, not permutations.

(8-choose-2) * (12-choose-1)
= 28 * 12
= 336
possible veg, veg, non-veg committees.

Likewise, if we have a veg, non-veg, non-veg committee, there are
(8-choose-1) * (12-choose-2)
= 8 * 66
= 528 possible committees.

Since both
veg, veg, non-veg
and
veg, non-veg, non-veg
are options, we must add the two possibilities together.

336 + 528 = 864 unique three-person committees that have at least one vegetarian and at least one non-vegetarian.

In choosing three pizza toppings out of 9, we will again assume that order doesn't matter. Putting pepperoncini on top of pineapple is not a different pizza than pineapple on top of pepperoncini.

So, we calculate the combinations for all pizzas up to three toppings:

(9-choose-3)
= 84 possible three-topping pizzas.

(9-choose-2)
= 36 possible two-topping pizzas

(9-choose-1)
= 9 possible one-topping pizzas

Adding these up, we get:
84 + 36 + 9 = 129 different combinations of up to 3 toppings, out of 9 total toppings.