You’ve got a coupon for a 12 topping pizza, and though you will have to double u

Question

You’ve got a coupon for a 12 topping pizza, and though you will have to double up (or more) on some of the topping you plan to get all 12 toppings out of it! The pizza place has the following 6 toppings: pepperoni, sausage, pineapple, green chile, onions, and mushrooms. How many different pizzas can you get if:

(a) you have no restrictrions on the number of each topping?

(b) you get at least two of each kind of topping?

(c) your roommate demands that you get at least 7 toppings worth of green chile and at least one of pineapple?

(d) your roommate demands that you get exactly 5 toppings worth of pepperoni?

Explanation / Answer

(a) 1) You can get 6 different toppings if you select 2 of each toppings or

2) you can get 1 topping if you select all 12 pizzas of one topping only or

3) you can get 5 different toppings if you select 4 pizzas of one topping and remaining 8 pizzas from 4 other toppings (2 from each) or

4) you can get 4 different toppings if you select 9 from one topping and remaining 3 pizzas from 3 other toppings or

5) 3 different toppings if selected 10 of one topping and remaining 2 of others or

6) 2 different toppings if selected 6 pizzas from any 2 toppings.

There are many such combinations or selections but all falling in any of the above 6 cases, for example, in 6th case above, you can select 11 pizzas of one topping and remaining 1 pizza from any other topping instead of 6 pizzas from 2 different toppings.

So, the answer is, you can get 1 or 2 or 3 or 4 or 5 or 6 different toppings when you don't have any restrictions.

(b) you get at least 2 pizzas of each topping. This means, 2 or more of each topping. 1) you can get 2 pizzas from each of 6 toppings. 2*6=12. So, you can get exactly 6 different toppings. Any other cases are not possible.

Answer is: exactly 6 different toppings.

(c) at least 7 of green Chile and at least 1 of pine apple. So, you can get in the following way:

7 of green Chile, 1 of pine apple and remaining 4 as below:

1) 4 of green Chile: so, 2 toppings

2) 4 of pine apple: so, 2 toppings

3) 4 of any other one: so, 3 toppings.

4) 2 from other 2: so, 4 toppings.

5) 1 & 3 or 3&1: so, again 4 toppings.

The answer is: 2 or 3 or 4 different toppings.

(d) exactly 5 of pepperoni, other 7 can be selected as below:

1) 7 from any other one topping from 5 toppings other than pepperconi. (2 toppings)

2) 1 from one topping and remaining 6 from one topping. (3 toppings)

3) 2 from one & 5 from one (3 toppings)

4) 3 from one & 4 from one (3 toppings)

5) one from one and remaining 6 from other 4 toppings as below:

--6 from one of those 4 toppings (3 toppings).

--3 & 3 from any 2 toppings (4 toppings)

-- 2 & 2 & 2 from any 3 toppings (5 toppings)

-- 2 & 2 & 1 & 1 from each of those toppings. (So, 6 toppings) and other such combinations.

Answer is: you can get 2 or 3 or 4 or 5 or 6 different toppings but not one because exactly 5 pizzas of pepperconi must be there and other 7 have to be taken from other 5 toppings.

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