Take the task of creating a meal and break it into stages.

Stage 1: Select 3 entrees
Since the order in which we select the entrees does not matter, we can use combinations.
We can select 3 entrees from 5 entrees in 5C3 ways (10 ways)
So, we can complete stage 1 in 10 ways

If anyone is interested, we have a video (below) on calculating combinations (like 5C3) in your head

ASIDE: We now have our 3 entrees, but we don't have a sauce for each entree.

Stage 2: Choose a sauce for one of of the entrees
We have 2 choices (garlic or chili sauce), so we can complete stage 2 in 2 ways

Stage 3: Choose a sauce for another entree
We have 2 choices (garlic or chili sauce), so we can complete stage 3 in 2 ways

Stage 4: Choose a sauce for the last remaining entree
We have 2 choices (garlic or chili sauce), so we can complete stage 4 in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create a meal) in (10)(2)(2)(2) ways (= 80 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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I would say it is a tricky question since it is ambiguous.
There are 2 sources of confusion:
1) Do we count chicken, prawn, beef, and chicken, beef, prawn as same or as a different way of serving? I guess we can conclude that in one way, so we will need divide by 3!
2) Do we serve a sauce on the entree? Can we order a garlic chicken and chili beef? So we need to multiply by 8? can we order chicken without sauce? We will need to multiply by 27. Do we get a sauce in Panda Express way, we will need to multiply by 2, 3 or 4 depends on whether we can reject both or take both.

Anyway, with all this in mind, we can calculate that we have 5*4*3=60/3! = 10 * 8 = 80
Since answer C
But if you will calculate an ordered permutation 60 and multiply by 2 Panda Express style, you will get incorrect 120

Can someone help solve this question with a clear method?

Hi,

I am really confused.

1> I first tried to answer assuming ordered permutation way and got 120.
2> If it is not ordered but rather a combination of 3 items (out of chicken/beef/prawn/veg/tofu) in one entry, then their should be 5C3 ways of combining them - 10 ways which then should be multiplied by 2 ways of adding sauce. So the answer was 20.

Can someone please clearly lay down the strategy if there are subtle nuances in the question?

could anyone explain the solution please? I cannot reach 80 anyhow.

thank you GMATPrepNow
this is hard to conceive, probably i've never faced this type of question

Why do you have so many stages? Why wouldn't it be just two stages? I thought of just 2, the entrees and the sauces. So it would be 20 in total, or 5C3 * 2.
Another way I thought of was if you cannot repeat an entree, there are 24 options (5*2 + 4*2 + 3*2).

Can you explain why both of these solutions are wrong?

It helps to avoid errors



What do your values stand for?

While my solution might LOOK longer than it needs to be, I don't write out every stage on my notepad. HOWEVER, I make it very clear to myself what each is occurring at each stage.
So, for example, in your 2-stage solution, 5C3 * 2, what does 5C3 represent and what does 2 represent?

Likewise, in your other solution, 5*2 + 4*2 + 3*2, what does each part represent?

Once you start articulating what each stage represents, your solutions will improve.

Cheers,
Brent
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It's a tricky question.
It's also poorly-worded.

Cheers,
Brent
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Okay, here you go...
1. Why do you have 4 stages rather than two? You have 4 stages for entree * sauce * sauce * sauce. Why? Why not two stages for Entrees and Sauces. With two stages you would do the exact same thing you did, but without the additional two stage. 5C3 (choose 3 entrees from five entrees) * 2 (sauces).
2. Why is it also not appropriate to calculate as Option 1 + Option 2 + Option 3? Option 1 = 5 entrees * 2 sauces. Option 2 = 4 entrees * 2 sauces. Option 3 = 3 entrees * 2 sauces.

It would help if you actually defined each stage.
For example:
STAGE 1: Choose 3 entrees
This stage can be completed in 5C3 (aka 10) ways

STAGE 2: What is happening here?
Is it "Choose a sauce"?
If so, which entree are you choosing a sauce for? Each entree must have its own accompanying sauce.
So, for example, ONE outcome (aka meal) might be: chicken with chili sauce, beef with garlic sauce, and vegetable with chili sauce



This looks like you are repeating many possible meals. So, your answer will be far too big.
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I appreciate your solution and it is well explained. Can you please help me understand what I am doing wrong in the following approach,

Stage 1: No of ways to make 1st entree (select 1 base from 5 and select 1 sauce from 2) = 5C1 x 2C1 = 10
Stage 2: No of ways to make 2nd entree (select 1 base from 4 remaining and select 1 sauce from 2) = 4C1 x 2C1 = 8
Stage 3: No of ways to make 3rd entree (select 1 base from 3 remaining and select 1 sauce from 2) = 3C1 x 2C1 = 6

Total number of ways = 10 x 8 x 6 = 480

The problem with your solution is that you have counted each unique meal 6 times.

In your solution, we COULD choose:
- chicken with garlic sauce in stage 1, prawn with chili sauce in stage 2, and beef with garlic sauce in stage 3
or
- chicken with garlic sauce in stage 1, beef with garlic sauce in stage 2, and prawn with chili sauce in stage 3
or
- prawn with chili sauce in stage 1, chicken with garlic sauce in stage 2, and beef with garlic sauce in stage 3
or
- prawn with chili sauce in stage 1, beef with garlic sauce in stage 2, and chicken with garlic sauce in stage 3
or
- beef with garlic sauce in stage 1, prawn with chili sauce in stage 2, and chicken with garlic sauce in stage 3
or
- beef with garlic sauce in stage 1, chicken with garlic sauce in stage 2, and prawn with chili sauce in stage 3

Notice that all 6 meals are the SAME.
Since you have counted each unique meal 6 times, we must divide your answer by 6 to get 460/6 = 80

Does that help?

Cheers,
Brent
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Hi Brent,
This does help. Thank you for clearing this up. I really appreciate it.
Best Regards,

Since you cannot choose 1 Entree from each kind - 5C3 = 10 - let's say we choose chicken, tofu and beef

Then for each of the 3 -
Chicken - 2 options of sauce
Beef - 2 options of sauce
Tofu - 2 options of sauce

Hence 10*2*2*2 - 80

C

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