An ice cream sundae at an ice cream shop consists of 3 different types of ice cream and 2 different types of topping. If there are 12 different types of ice cream and 8 different toppings to choose from, how many different sundae's can a customer create?
My approach: $$_{12}C_3 * {_8}C_2=6160$$
Is this correct?
Best Answer
Yes, your approach is correct.
Let me explain:
Since ice-cream and topping are independent, $$#of sundays=#of icecreams * #of toppings$$
If types of ice-cream didn't have to be different your result would be $12^3$
Now you have to choose one flavor (12 ways), then a different flavor (11 ways) and then a third one (10 ways). So your result would be $12*11*10$.
But keep in mind that you are not interested in the order of the flavors. There are $3!$ ways of ordering the 3 flavors. So, your true number of different ice-creams is$\frac{12*11*10}{3!}$ which is $_{12}C_3$.
Same way for toppings.
If you define as different sundae a sundae where the order of the ice-creams or the order of the toppings is different, then your result would be $(12*11*10)*(8*7)$