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# Permutations

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In how many permutations of the word "ROLLER" can the 4 letters "LLRR" be together? And in how many permutations can the letters "ROR" be together? Thanks for help.

Jul 2, 2020

#1
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There are four letters in LLRR, so there would be 4! possible different arrangements. However, there are two repeating letters, so we have to divide by 2!2! to account for overcounting: 4!/2!2! = 4*3*2/4 = 6, so there are six possible different arrangements of the word LLRR.

(If you're asking for a ratio of the permutations of LLRR vs the permutations of ROLLER, then do 6/(6!/6), where 6!/6 accounts for the arrangements of the word ROLLER and 6 accounts for the arrangements of the word LLRR.)

There are three letters in ROR, so there would be 3! possible different arrangements. However, there is one repeating letter, so we have to divide by 2! to account for overcounting: 3!/2! = 6/2 = 3, so there are three possible different arrangements of the word ROR.

(If you're asking for a ratio of the permutations of ROR vs the permutations of ROLLER, then do 3/(6!/6), where 6!6 accounts for the arrangements of the word ROLLER and 3 accounts for the arrangements of the word ROR.)

I hope I helped!

Jul 2, 2020
#2
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Pretty good job young person! The only mistake you made was in the letters "ROR". Each permutation begins with one of the 4 letters:E-L-O-R. Beginning with E, "ROR" should appear 3 times with the letters E-L-L.[ELLROR, ELRORL, ERORLL]. Beginning with L, it should appear another 3 times, but because we have 2 L's, it should appear a total of 6 times.  [LELROR,  LLRORE,  LROREL, LERORL, LLRORE, LLEROR].          Beginning with letter "O" is a non-starter, because it sits smack in the middle of "ROR". That leaves the letter R itself. Beginning with R, we should have another 3 combinations with E-L-L. [RORELL, RORLEL, RORLLE].
So, in summary, we should have 12 "ROR" permutations in total.

Jul 2, 2020