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Combinatorics - Restaurant Menu!!

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by kunalkulkarni » Mon Feb 11, 2013 8:49 am
Hi,
Most of us know how to solve the below problem:

A restaurant menu features five appetizers, six entrees, and three desserts. If a dinner special consists of one appetizer, one entree, and one dessert, how many different dinner specials are possible? (5*6*3 = 90 different ways)

I was curious to know, how to solve the same problem with a twist:
A restaurant menu features five appetizers, six entrees, and three desserts. If a dinner special consists of two appetizers, two entrees, and two desserts, how many different dinner specials are possible?

Can someone please explain me the concept for above? It will help me to create a good foundation for the combinatorics problems.

Thanks in advance,
Kunal
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by Brent@GMATPrepNow » Mon Feb 11, 2013 8:56 am
kunalkulkarni wrote: A restaurant menu features five appetizers, six entrees, and three desserts. If a dinner special consists of two appetizers, two entrees, and two desserts, how many different dinner specials are possible?
Take the task of "building" a meal and break it into stages.

Stage 1: Select 2 appetizers.
Since the order of the selected appetizers does not matter, we can use combinations here.
We can select 2 appetizers from 5 appetizers in 5C2 ways (10 ways).

Aside: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Stage 2: Select 2 entrees.
Once again, the order of the selected entrees does not matter, so we can use combinations.
We can select 2 entrees from 6 entrees in 6C2 ways (15 ways).

Stage 3: Select 2 desserts.
We can select 2 desserts from 3 desserts in 3C2 ways (3 ways).

By the Fundamental Counting Principle (FCP) we can complete all 3 stages (and thus create a meal) in (10)(15)(3) ways ([spoiler]= 450 ways[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775
Brent Hanneson - Creator of GMATPrepNow.com
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by kunalkulkarni » Mon Feb 11, 2013 9:07 am
Thank you Brent. Appreciate your quick response.
Your answer reinforced my concept and now I know, I am thinking in right direction.
I will also check out your 'aside-notes' to make the concept crystal clear.

Cheers,
Kunal
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