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hard counting

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137
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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?

Dec 3, 2022

I uh think that the answer to (a) is $$2^{10}$$