Veritas problem--Combinatorics

dddanny2006 wrote:Hi

Please tell me where Im going wrong here.

Here's my method to it


Since we have 3 different burgers to choose from,3 different drinks to choose from and 2 different side to choose from

We use the slot method

3*3*2
3!

which equals 3combinations.Here we divide by 3 factorial because order doesnt matter,we can choose
the drink first,the burger second and sides third ,or any another way and the order doesnt matter.
Why am I wrong?

GMATGuruNY wrote:

dddanny2006 wrote:Hi

Please tell me where Im going wrong here.

Here's my method to it


Since we have 3 different burgers to choose from,3 different drinks to choose from and 2 different side to choose from

We use the slot method

3*3*2
3!

which equals 3combinations.Here we divide by 3 factorial because order doesnt matter,we can choose
the drink first,the burger second and sides third ,or any another way and the order doesnt matter.
Why am I wrong?

I think of this as a problem with MULTIPLE BUCKETS.
We have a bucket of BURGERS, a bucket of DRINKS, and a bucket of SIDES.
We have to choose from each bucket.
Given a problem with multiple buckets:

1. Count the number of options from each bucket.
2. Multiply the results.

Bucket of burgers --> 3 options.
Bucket of drinks --> 3 options.
Bucket of sides --> 2 options.

To combine the options from each bucket, we multiply:
3*3*2 = 18.

Other problems with multiple buckets:

https://www.beatthegmat.com/combinations ... 91440.html
https://www.beatthegmat.com/combination- ... 85034.html
https://www.beatthegmat.com/combinatorics-t71136.html

Brent@GMATPrepNow wrote:

dddanny2006 wrote:Thanks.Why am I wrong?Why is my understanding letting me down? The order Drink,burger and sides are the same as Side,drink,burger.That's why I divided by the number of interchangeable slots.Please explain.

This question is similar to the "does order matter" question frequently asked by students preparing to tackle counting questions. If you're interested, I wrote an article for BTG about it: https://www.beatthegmat.com/mba/2013/09/ ... s-part-iii

When applying the Fundamental Counting Principle (aka Slot Method), it's important to define what each stage (slot) represents. Once you've done so, you should ask, “Does the outcome of each stage differ from the outcomes of the other stages?”

If the answer to this question is YES, then we can continue solving the question using the Fundamental Counting Principle. If the answer to this question is NO, we cannot solve the question using the Fundamental Counting Principle and, in most cases, we can use combinations.

So, for this question, we can take the task of building a meal and break it into stages:
Stage 1: Select a burger
Stage 2: Select a drink
Stage 3: Select a side

Does the outcome of each stage differ from the outcomes of the other stages?
Yes. For example, the outcome of stage 1 is selecting a burger. This is different from the outcome of stage 2, which is selecting a drink.

Since the three outcomes are all different, we can apply the Fundamental Counting Principle (FCP).

That is, . . .
Stage 1: Can be completed in 3 ways
Stage 2: Can be completed in 3 ways
Stage 3: Can be completed in 2 ways

So, by the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus build a meal) in (3)(3)(2) ways ([spoiler]= 18 ways[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Brent@GMATPrepNow wrote:

dddanny2006 wrote:Thanks Brent.So if I got to check whether each slot is filled from the selection pool.If its a No then the FCP works,if its a Yes we got to use combinations.Correct?

I'm not sure what you mean by "selection pool," so I don't want to provide an answer that leads you astray.
I prefer to ask, “Does the outcome of each stage differ from the outcomes of the other stages?”


Cheers,
Brent

Brent@GMATPrepNow wrote:

dddanny2006 wrote:By selection pool I meant pool of burgers,pool of drinks and pool of sides.

Brent@GMATPrepNow wrote:

dddanny2006 wrote:Thanks Brent.So if I got to check whether each slot is filled from the selection pool.If its a No then the FCP works,if its a Yes we got to use combinations.Correct?

I'm not sure what you mean by "selection pool," so I don't want to provide an answer that leads you astray.
I prefer to ask, “Does the outcome of each stage differ from the outcomes of the other stages?”


Cheers,
Brent

I'm still not sure what you mean (although it's entirely possible that your rationale is perfect).

Consider the following example (classic combination question):
We want to create a 2-person committee from 5 people.
Let's start by trying to apply the FCP. So, we'll take the task of building the committee and break it into stages:
Stage 1: Select someone to be on the committee
Stage 2: Select someone else to be on the committee

Now ask, “Does the outcome of each stage differ from the outcomes of the other stages?”
The answer is NO. The outcomes are identical. In each case, the selected person is on the committee. As such, we cannot use the FCP.

Instead, we can use combinations. We can select 2 people from 5 people in 5C2 ways (10 ways)

How does this fit in with your question regarding "pools"?

Cheers,
Brent