Combination/ Permutation with Duplicates

Question

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 - BO 1  and BO 2 ....I have written down the list of permutations and crossed out the duplicates.

BO 1 
 BO 2  
BK

O 1 B
O 1 O 2 
O 1 K

O 2 B 
 O 2 O 1  
 O 2 K

KB
KO 1 
 KO 2

Any help appreciated!

DonQuixote · Answer

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!.

KarishmaB · Answer

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

Combination/ Permutation with Duplicates

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