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Can I ask why the following is wrong?

Total possible blends = 4! = 24
Blends with 1 ingredient = 4!/3! = 4
Blends with 2 or more ingredients = 24-4 = 20

Why is the answer 11 and not 20? I understand the method of adding up blends with 2,3,4 ingredients, but why is my method wrong?

Thank you!

Hi strawberrycupcake,

Your first calculation (4!) is the total number of ways to ARRANGE ALL 4 ingredients (meaning that all 4 are used). However, that is not what the question asks us for - it's asking for the number of combinations of AT LEAST 2 of the ingredients.

Besides using 'brute force' to just list out all of the possibilities, you could use the Combination Formula 3 times and then add up the individual results:

4c2 + 4c3 + 4c4 = 6 + 4 + 1 = 11

GMAT assassins aren't born, they're made,
Rich

Hi Bunuel, why are we adding the number of bins and not multiplying it?

Hi siddharthsinha123,

We're asked the total number of bins, but the bins do NOT all contain the same number of ingredients. As such, while we can use the Combination formula to find the number of options for 2 ingredients, 3 ingredients and 4 ingredients, we're ultimately adding up 3 different sub-totals... so we have to use addition (and not multiplication) to get to that total.

GMAT assassins aren't born, they're made,
Rich

Hi,

I would like to ask which part of the problem indicates, that each ingredient can be used only once in the mix.

The only part that could do this is last sentence in brackets, but I can't really get a grasp on it.

And if there is no such assumption we have an infinite number of possible answers.

Hi ascone,

When working through a PS question on the GMAT, if you interpret the question a certain way and find that "your answer" is not among the 5 choices, then you have to reconsider how you interpret the question.

This prompt does NOT say anything about the possible 'ratios' of the ingredients in the blends, it just asks for the number of possible blends that use 2 (or more) ingredients.

For example:
-If one blend uses ingredients A, B and C, then that is DIFFERENT from one that uses just A and B.

-If a blend uses '1 part A' and '1 part B', then that mixture is the SAME as one that uses '2 parts A and 1 part B.'

Thinking in those terms, can you determine all of the possible blends?

GMAT assassins aren't born, they're made,
Rich

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