Problem Solving

Create your own sundae:
12 ice Cream Flavors
10 Kinds of Candies
8 Liquid Toppings
5 Kinds of Nuts
With or Without Whipped Cream

If a customer makes exactly one selection from each of the five categories shown in the above table, what is the greatest possible number of different ice cream sundaes that a customer can create?


I know it is a combination problem, 5 from 37. But how to ensure 1 from 12, 1 from 10, 1 from 8, 1 from 5 and 1 out of 2??

cnseetharaman wrote:Create your own sundae:
12 ice Cream Flavors
10 Kinds of Candies
8 Liquid Toppings
5 Kinds of Nuts
With or Without Whipped Cream

If a customer makes exactly one selection from each of the five categories shown in the above table, what is the greatest possible number of different ice cream sundaes that a customer can create?


I know it is a combination problem, 5 from 37. But how to ensure 1 from 12, 1 from 10, 1 from 8, 1 from 5 and 1 out of 2??

Hi!

The question is actually much easier than you think. Let's start with a simpler version of the same question:

John is going to pick one appetizer and one main course for his meal. If the only appetizers available are salad and soup and the only main courses are fish, beef and chicken, how many different meals could John choose?

Solving by brute force, we get:

Salad/Fish
Salad/Chicken
Salad/Beef
Soup/Fish
Soup/Chicken
Soup/Beef

for a total of 6 possible meals.

However, what we're really doing is picking one selection out of the two possibilities for each course. So, in terms of combinatorics we have:

Appetizers: 2C1 = 2
Main Courses: 3C1 = 3

Here's one of the most important thing to remember about combinations, permutations and probability:

if you're counting MULTIPLE possibilities, MULTIPLY the individual possibilities;

and

if you're counting ALTERNATIVE possibilities, ADD the individual possibilities.

Here, we're choosing one appetizer AND one main course, so we MULTIPLY:

2C1 * 3C1 = 2*3 = 6

Now, back to your question:

Each sundae consists of one flavour AND one candy AND one liquid AND one nut AND with/without whipped cream. So, we count the number of possibilities from each category and MULTIPLY them together.

Flavour: 12C1 = 12
Candy: 10C1 = 10
Liquid: 8C1 = 8
Nuts: 5C1 = 5
With/without whipped cream: 2C1 = 2

Accordingly, we have 12*10*8*5*2 = 120*8*10 = 1200*8 = 9600 different possible sundaes!

If you're not sure what exactly "12C1" (read as "twelve choose one") means, then you need to brush up on your combinatronics - but you really don't need to know any fancy math to realize that if there are 12 options and you're going to choose exactly 1 of them there are 12 possible choices.

cnseetharaman wrote:Create your own sundae:
12 ice Cream Flavors
10 Kinds of Candies
8 Liquid Toppings
5 Kinds of Nuts
With or Without Whipped Cream

If a customer makes exactly one selection from each of the five categories shown in the above table, what is the greatest possible number of different ice cream sundaes that a customer can create?


I know it is a combination problem, 5 from 37. But how to ensure 1 from 12, 1 from 10, 1 from 8, 1 from 5 and 1 out of 2??

The number of ways he can choose 1 Ice cream flaovor is = 12C1 = 12.
So, it boils down to a multiplication of all the categories.

Number of Different Ice Cream Sundaes = 12*10*8*5*2 = 9600.