the ice cream parlor offers 15 ice cream flavors and the following toppings: 5 t

Question

the ice cream parlor offers 15 ice cream flavors and the following toppings: 5 types of sprinkles, 10 types of fruits, and 7 types of special candies. how many different ways are there to order an ice cream dessert, if the customer must choose an ice cream flavor, but need not choose any of the other offerings? ( note: a customer can only choose one of each topping; e.g.if the set of fruit toppings are given by F={F1,F2...,apple,orange...,F10}, a customer can't choose apple twice, but he can choose apple and orange).  

Please explain this thoroughly :)

Explanation / Answer

lets take a simple example first. You have the choice of apple and Orange. How many ways are there where you have to choose something (something includes nothing as well). You can choose (1) Nothing (2) Apple (3) Orange and (4) Apple and Orange. So you have 4 options.

In general when you have to choose something out of n given options, the number of ways = 2n, here n= 2, therefore 22 = 4.

Lets take A, B, C- Then (1) Nothing (2) A (3) B (4) C (5) A and B (6) A and C (7) B and C and (8) A, B and C

which is 23 = 8.

Now, if you had to choose at least 1, it means he option of choosing nothing is out. Then the total number of ways of choosing at least 1 out of n is 2n - 1 (as we are excluding nothing)

Now he can choose 1 out of any 15 ice cream flavours in 15 ways (if he was allowed to choose multiple ice cream flavours and since he has to choose at least 1, it would have been 215 - 1), but the question says 'an' ice cream flavour, which means 1.

So for sprinkles = 25, Fruits = 210 and candies = 27 as you have to choose something, where 0 is included.

Therefore Total number of ways = 15 * 25 * 210 * 27 = 15 * 32 * 1024 * 128 = 62914560 ways.

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