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The Natural Woman, a women's health food store, offers its

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The Natural Woman, a women's health food store, offers its  [#permalink]

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Updated on: 27 Aug 2014, 04:31
6
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Difficulty:

35% (medium)

Question Stats:

73% (01:47) correct 27% (02:03) wrong based on 176 sessions

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The Natural Woman, a women's health food store, offers its own blends of trail mix. If the store uses 4 different ingredients, how many bins will it need to hold every possible blend, assuming that each blend must have at least two ingredients? (Also assume that each bin can hold one and only one blend.)

(A) 1
(B) 4
(C) 7
(D) 11
(E) 10

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Originally posted by royQV on 27 Aug 2014, 01:31.
Last edited by Bunuel on 27 Aug 2014, 04:31, edited 1 time in total.
Renamed the topic and edited the question.
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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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27 Aug 2014, 04:34
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2
royQV wrote:
The Natural Woman, a women's health food store, offers its own blends of trail mix. If the store uses 4 different ingredients, how many bins will it need to hold every possible blend, assuming that each blend must have at least two ingredients? (Also assume that each bin can hold one and only one blend.)

(A) 1
(B) 4
(C) 7
(D) 11
(E) 10

The number of bins with 2 different ingredients = $$C^2_4=6$$;
The number of bins with 3 different ingredients = $$C^3_4=4$$;
The number of bins with 4 different ingredients = 1 ($$C^4_4=1$$).

The total number of bins with at least two ingredients = 6 + 4 + 1 = 11.

Hope it's clear.
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17 May 2015, 14:02
1
Hi All,

The answers to this question are relatively small, so if you don't see the "math" approach to solving this question, then you can 'brute force' the solution by "mapping out" all of the possibilities...

We're told that there are 4 ingredients (we'll call them A, B, C and D); we're told that each 'mix' must include AT LEAST 2 ingredients...

2 ingredient blends:
AB
AC
BC
BD
CD

3 ingredient blends:
ABC
ABD
ACD
BCD

4 ingredient blends:
ABCD

Total blends = 6+4+1 = 11

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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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30 May 2018, 07:59
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!
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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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30 May 2018, 12:54
1
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

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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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05 Mar 2019, 15:45
1
Bunuel wrote:
royQV wrote:
The Natural Woman, a women's health food store, offers its own blends of trail mix. If the store uses 4 different ingredients, how many bins will it need to hold every possible blend, assuming that each blend must have at least two ingredients? (Also assume that each bin can hold one and only one blend.)

(A) 1
(B) 4
(C) 7
(D) 11
(E) 10

The number of bins with 2 different ingredients = $$C^2_4=6$$;
The number of bins with 3 different ingredients = $$C^3_4=4$$;
The number of bins with 4 different ingredients = 1 ($$C^4_4=1$$).

The total number of bins with at least two ingredients = 6 + 4 + 1 = 11.

Hope it's clear.

Hi Bunuel, why are we adding the number of bins and not multiplying it?
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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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05 Mar 2019, 15:55
1
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.

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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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12 Mar 2019, 00:46
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.
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Re: The Natural Woman, a women's health food store, offers its  [#permalink]

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12 Mar 2019, 11:35
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?

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Rich
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Re: The Natural Woman, a women's health food store, offers its   [#permalink] 12 Mar 2019, 11:35
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