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a) Grogg is ordering pizza from a local pizzeria, which offers ten different toppings: pepperoni, mushrooms, sausage, onion, olives, green peppers, pineapple, spinach, garlic, and hummus (an \(\text{Ao} \text{PS}\) favorite).  For any pizza, any combination of toppings is possible, including no toppings.  How many different pizzas can Grogg order, if Grogg must order at least five toppings?

(b) Lizzie hears about Grogg's plans, and is also interested in ordering a pizza.  For variety, they agree to order one pizza each so that they have no toppings in common.  In how many different ways can Lizzie and Grogg order their pizzas, if each pizza must have at least three toppings?

(c) It turns out Winnie is also interested in ordering a pizza!  Lizzie and Grogg still agree that their pizzas can't have any toppings in common, but Winnie will include a topping on her pizza only if it appears on Grogg's pizza or Lizzie's pizza (but she doesn't have to include it).  In how many different ways can Grogg, Lizzie, and Winnie order their pizzas, if Lizzie's pizza must have at least four toppings, and Grogg's pizza must have at least one topping?

You may leave your answers in exponential form.

 Nov 12, 2022
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I am seriously saying that this seems like a mess, though not trying to be rude.

 

(a) Can those five toppings even be the same or do they have to be different from each other? Maybe I am currently misunderstanding the question, just asking.

 

Since no one said that they must be different, which to be honest does not make sense, would be 'horrid' and unlogical.

 

assume them to be different from each other:

 

If that is the case, topping 1 can be any of these 10 ingredients, topping 2 one less, topping 3 one less from topping two and so on. Now think about what to do with these possibilities and what truly makes sense to you.

 

I may come back if the question aforementioned is answered.

 Nov 12, 2022

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