August 19th, 2007, 02:23 AM  #1 
Newbie Joined: Aug 2007 From: Bangladesh Posts: 1 Thanks: 0  Permutation problem
Q: In how many ways the letters in the word PARALLEL can be arranged so that there will always be two "L"s together as in the original word? And my solution is : 7!/2!6!/2! PARALLEL has 8 letters with one letter appearing 3 times and 1 letter appearing 2 times. The letters can be arranged in 8!/(3!*2!) = 3360 ways. If you want 2 Ls together, you can treat it is 1 single letter. So it seems that the letters can be arranged in 7!/2!(= 2520) ways. but when 3 Ls are together we can't distinguish between the pair LL and lone L. So permutations 7!/2! (= 2520) include the permutation like "PAR(LL)LAE" & "PARL(LL)AE". This problem arise when 3 Ls are together. SO In how many permutations are they together? In 6!/2! permutations 3 Ls are together. But when you disregard the occasional identical character between LL and L, the number of times 3Ls apear together is doubled (we have disregarded this character to get 7!/2!). So 2*6!/2! number times 3 Ls have appeared together in our permutation (7!/2!), half of them are identical. So the number of valid permutation is 7!/2!6!/2! = 2160 I will ask for vote. Whether you think I'm correct or wrong! 
August 19th, 2007, 01:45 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,757 Thanks: 2138 
There are 6 positions for "LLL", 2×5 positions for "LL" at one end and another "L" separate from the "LL", and 5×4 position for "LL" elsewhere and another "L" separate from the "LL". That's a total of 36 ways of placing the "L"s with at least two together somewhere. For each of those ways, the "A"s can be added in 5×4/2! ways, i.e., 10 ways, and then the "P", "R" and "E" can be added in 3! ways, i.e., 6 ways. Hence the required total is 36×10×6, i.e., 2160.

August 20th, 2007, 08:14 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Yes, you're both right.


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