This question has two parts I did the first part Ill show my

This question has two parts, I did the first part (I\'ll show my solution) but for the second part I\'m not sure how to do the counting. Anyways, here\'s the question:

4. a) Homer’s Pizza is advertising the following deal:
• 2 pizzas
• up to 3 toppings on each pizza
• 7 toppings to choose from
• total cost $10.99

The pizza toppings must be unique—double or triple toppings are not allowed. Two of the same pizza is allowed. The arrangement of the toppings on each pizza does not matter; e.g., tomatoes on top of pepperoni is the same as pepperoni on top of tomatoes. What is the total number of possibilities for a pizza order in this deal?

Now for part b:

Repeat part a, but this time when double or triple toppings are allowed; i.e., 2 or 3 of the toppings can be the same. (Note that double or triple toppings are not required, but they are allowed.)

The last person who helped me gave me a complete wrong answer which makes no sense. Intuitively, and I actually wrote down all the possibilities manually using excel to see how to do this, there are only 7 ways to get 1 pizza with the same 3 topings, if the topings are A, B, C, D, E, F, G; AAA, BBB, CCC, DDD, EEE, FFF, GGG. And the person who gave me an answer said 11, which makes no sense. Now I actually wrote down all the possibilities and I found that indeed there are only 7 ways to get get three topings on one pizza and I also found that there\'s an additional 56 ways to get the same topings, 49 from 3 toping pizzas and 7 from pizzas with only two topings.

Solution

Solution Part a)

The number of ways that one pizza can be made. Our pizza may have zero, one, two, three toppings, so the total number of pizzas available to us is:(Out of 7 Toppings)

7C0+7C1+7C2+7C3=1+7+21+35=64

Thus, if we are to choose two pizzas

But Both pizzas are different. In this case, there are 64 pizzas to choose from and I want to pick two, so the number of ways to do this is:

64C2=2016

So the total number of ways to pick the two pizzas is:

64+2016=2080

Answer for Part a is 2080 ways of Possibilities.

Solution Part B:

Suppose We have 7 toppings A B C D E F G

Just mark an X over the symbol for each topping you order. If you wanted double pepperoni, two X\'s would be placed in the P slot, counting as two toppings.

If we allow multiple toppings, we can calculate the total number of pizzas possible by considering the total number of ways that the order form can be filled out.

Thus, each 3 item pizza will be represented by an arrangement of 3 X\'s and 6 lines. The number of ways to arrange these 9 symbols is:

9P3=9!/(3!*6!)=84

To account for all possible pizza choices, we need to consider pizzas with zero, one, two, three toppings. If we do this we get:

9P3+8P2+7P1+6P0=[9!/(3!*6!)]+[8!/(2!*6!)]+[7!/(1!*6!)]+[6!/(0!*6!)]

=84+28+7+1

=120

So, the number of choices for two pizzas is then 120+120C2= 120+7140=7260

Answer for Part b is 7260 ways of Possibilities.