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# Combination/ Permutation with Duplicates

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Intern
Joined: 07 Jan 2012
Posts: 8

Kudos [?]: 6 [0], given: 0

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14 Aug 2012, 14:02
Hi all,

This is my first post here, so I apologise if this has been asked.

How many unique ways can you re-arrange the word BOOK if you only select 2 letters?

when I try and do this, the total number of permutations is 12 (4P2)...However, I am not sure how to mathematically calculate the number of duplicates (in this case, there are 5 duplicates)? Ie - BO1 and BO2....I have written down the list of permutations and crossed out the duplicates.

BO1
BO2
BK

O1B
O1O2
O1K

O2B
O2O1
O2K

KB
KO1
KO2

Any help appreciated!

Kudos [?]: 6 [0], given: 0

Manager
Joined: 25 Jun 2012
Posts: 56

Kudos [?]: 23 [0], given: 2

Re: Combination/ Permutation with Duplicates [#permalink]

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16 Aug 2012, 20:38
In this question, you divide the 12 possible permutations by 2!, as one of the letters (O) appears 2 times. Note that O1O2 doesn't work, because that gives you the same words. Total number of permutations is therefore 6, or 4P2 / 2!.

Kudos [?]: 23 [0], given: 2

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7676

Kudos [?]: 17381 [0], given: 232

Location: Pune, India
Re: Combination/ Permutation with Duplicates [#permalink]

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16 Aug 2012, 22:49
ashsundaresan wrote:
Hi all,

This is my first post here, so I apologise if this has been asked.

How many unique ways can you re-arrange the word BOOK if you only select 2 letters?

when I try and do this, the total number of permutations is 12 (4P2)...However, I am not sure how to mathematically calculate the number of duplicates (in this case, there are 5 duplicates)? Ie - BO1 and BO2....I have written down the list of permutations and crossed out the duplicates.

BO1
BO2
BK

O1B
O1O2
O1K

O2B
O2O1
O2K

KB
KO1
KO2

Any help appreciated!

The wording of the question is a little convoluted. " re-arrange the word BOOK if you only select 2 letters" is unclear. I think they mean "how many 2 letter words can you make from the letters of the word BOOK"

Out of the letters of the word BOOK, you need to make distinct 2 letter words.
The two letters can be same or they can be different.
Two letters same - This can be done in only 1 way i.e. by selecting both Os. The word will be OO.
Two letters different - Out of the 3 distinct letters (B, O, K), select any two and arrange them in 3p2 = 6 ways.
Total number of distinct 2 letter words = 1+6 = 7
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Kudos [?]: 17381 [0], given: 232

Re: Combination/ Permutation with Duplicates   [#permalink] 16 Aug 2012, 22:49
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