himurakenshin said:
pizza store has small/medium/large with 10 different toppings 2 crusts and 3 types of sauses. how many ways to ordera pizza with atleast 1 topping and 1 sauce?
based on previous thread contributions:
# orders for each of 3 sizes, each of 2 crusts, at least 1 of 10 toppings, and at least 1 of 3 sauces =
(3 sizes)
⊗(2 crusts)
⊗
(number ways choosing 1 topping from 10
+ number ways choosing 2 toppings from 10
+ number ways choosing 3 toppings from 10
+ ... ... ... number ways choosing 10 toppings from 10
)
⊗
(number ways choosing 1 sauce from 3
+ number ways choosing 2 sauces from 3
+ number ways choosing 3 sauces from 3
)
= \ 6 \cdot \left ( \sum_{r=1}^{10} \mathbb{C}_{r}^{10} \right ) \cdot \left ( \sum_{r=1}^{3} \mathbb{C}_{r}^{3} \right )
= \ 6 \cdot \left ( \, (-1) \ + \ \sum_{r=0}^{10} \mathbb{C}_{r}^{10} \right ) \cdot \left ( \, (-1) \ + \ \sum_{r=0}^{3} \mathbb{C}_{r}^{3} \right )
= \ 6 \cdot \left ( \ (-1) \, + \, (2^{10}) \ \right ) \cdot \left ( \ (-1) \, + \, (2^{3}) \ \right )
= \ 6 \cdot \left (1023 \right ) \cdot \left (7 \right )
= \ 42,966
