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Amy and Adam are making boxes of truffles to give out as wed

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Amy and Adam are making boxes of truffles to give out as wed [#permalink]

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20 Jun 2011, 20:16
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Amy and Adam are making boxes of truffles to give out as wedding favors. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make?

I understand these are all the possible combinations:

Box 1: A B
Box 2: A C
Box 3: A D
Box 4: A E
Box 5: B C
Box 6: B D
Box 7: B E
Box 8: C D
Box 9: C E
Box 10: D E

But I don't get the mathematical interpretation of this.. 5! / (3! * 2!)

I don't get how the anagram can be "YYNNN"
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Re: Amy and Adam are making boxes of truffles to give out as wed [#permalink]

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20 Jul 2013, 12:14
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kiranck007 wrote:
Hi, I have an issue with comprehending such problems. Could any of you correct me please. I know the solution to the problem is 10.

My approach is : Each box can contain two truffles, so if I say there are two slots in the box then the first slot has the possibility of 5 different types of truffles and the next slot has the possibility of 4 different types of truffles. So i concluded 5*4 = 20 as the answer.

Can any of you please identify the flaw in my approach to the problem. Again, I know the solution, but I want to know what exactly is the flaw in the argument presented above, so that I don't repeat the mistake again.

The problem with your solution is that 20 has duplications in it: box holding {A, B} is the same as the box holding {B, A} and 5*4 will give you both boxes.

The question basically asks: in how many ways we can select 2 different truffles out of 5, which is simply $$C^2_5=10$$.

Hope it's clear.
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Re: Amy and Adam are making boxes of truffles to give out as wed [#permalink]

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20 Jul 2013, 12:17
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kiranck007 wrote:
Hi, I have an issue with comprehending such problems. Could any of you correct me please. I know the solution to the problem is 10.

My approach is : Each box can contain two truffles, so if I say there are two slots in the box then the first slot has the possibility of 5 different types of truffles and the next slot has the possibility of 4 different types of truffles. So i concluded 5*4 = 20 as the answer.

Can any of you please identify the flaw in my approach to the problem. Again, I know the solution, but I want to know what exactly is the flaw in the argument presented above, so that I don't repeat the mistake again.

I think you are using the slot-method. I am not very familiar with it, but you are missing the last step:

_ _ , two slots that can be filled in 5*4=20 ways. Correct till here, but then you have to divide for the "factorial" number of interchangeable slots, in this case $$2!=2$$. So the final result is $$\frac{5*4}{2}=10$$.
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Re: MGMAT Word Translations COMBINATORICS!!! [#permalink]

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21 Jun 2011, 02:39
Mahtab wrote:
Amy and Adam are making boxes of truffles to give out as wedding favors. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make?

I understand these are all the possible combinations:

Box 1: A B
Box 2: A C
Box 3: A D
Box 4: A E
Box 5: B C
Box 6: B D
Box 7: B E
Box 8: C D
Box 9: C E
Box 10: D E

But I don't get the mathematical interpretation of this.. 5! / (3! * 2!)

I don't get how the anagram can be "YYNNN"

You can think of it in 2 ways:
1. You have 5 different types of truffles. For each box, you have to select 2 different types of truffles. In how many ways can you do it?
5C2 = 5!/3!*2!
(When out of n things, you have to select r, you can do it in nCr ways. nCr = n!/r!*(n-r)!
For more on this, check out Combinatorics & Probability by Veritas Prep)
2. There are 5 truffles. You need to select 2 and not select the rest of the three. Y - select a truffle. N - Do not select it
So you can select 2 truffles in the number of ways in which you can arrange YYNNN.
YYNNN means you select the first 2 and leave the rest.
YNYNN means you select 1st and 3rd and leave the rest.
etc...
This can be done in 5!/2!*3!
You can arrange 5 things in 5! ways. But you have 2 Ys and 3 Ns so you divide by 2! and 3!.
(Again, for more on this, check out Combinatorics & Probability by Veritas Prep)
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Re: MGMAT Word Translations COMBINATORICS!!! [#permalink]

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21 Jun 2011, 05:37
Yeah I was doing so well in Quant up until I hit Guide 4 of MGMAT Combinatorics...they did an adaquete job at explaining it so I bought the Veritas Combinatorics book from amazon last night! I will put this to rest for a few days I guess and move on to statistics.

Thanks and I kind of understand the nCr thing but I want to go more in depth of the concept so I have a strong foundation.

Thanks!
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Re: MGMAT Word Translations COMBINATORICS!!! [#permalink]

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21 Jun 2011, 14:34
Project GMAT - also Veritas covers these fairly well, also

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Re: Amy and Adam are making boxes of truffles to give out as wed [#permalink]

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20 Jul 2013, 11:49
Hi, I have an issue with comprehending such problems. Could any of you correct me please. I know the solution to the problem is 10.

My approach is : Each box can contain two truffles, so if I say there are two slots in the box then the first slot has the possibility of 5 different types of truffles and the next slot has the possibility of 4 different types of truffles. So i concluded 5*4 = 20 as the answer.

Can any of you please identify the flaw in my approach to the problem. Again, I know the solution, but I want to know what exactly is the flaw in the argument presented above, so that I don't repeat the mistake again.

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Re: Amy and Adam are making boxes of truffles to give out as wed [#permalink]

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09 Jul 2014, 15:21
Can someone explain the differences in the approaches you would take if you could pick 2 of the same type of truffle vs how in this problem you can only use one type of each truffle per box?

I understand with the anagram method the latter is YYNNN - out of five choices, two of those five will be picked and three of those 5 will not be picked, therefore the answer is 5!/2!3!

But what about if dupes were allowed, how would the calculation change? Obviously that's a pretty easy thing to manually add but I'd like to understand the process.

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Re: Amy and Adam are making boxes of truffles to give out as wed   [#permalink] 09 Jul 2014, 15:21
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