Palatable Pizzeria Pizzas

In America, a pizza is ordered by choosing different kinds of toppings that are placed on the pizza crust. A newly opened pizzeria is running a promotion to get people interested in their pizzas and 7 different toppings. The promotion is that if you buy three large pizzas so that no pair of pizzas have any toppings in common, and every topping that they offer is on one of your pizzas, you get all three pizzas for only \(\$20\). How many different sets of 3 large pizzas actually qualify for the promotion?

Details and assumptions

Some of the pizzas could have no toppings.

The pizzas are only distinguishable by their toppings.

If the toppings are bacon, ham, chicken, beef, onions, mushrooms, pineapples, then a valid order is 1 pizza with bacon + onions + pineapples, 1 pizza with chicken + mushrooms and 1 pizza with ham + beef.


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