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 March 8th, 2015, 06:13 AM #1 Senior Member   Joined: Jan 2013 From: Italy Posts: 154 Thanks: 7 Combinatorics. How to select spaces for letters. Hi, I have the following problem: In how many ways can you arrange the letters of the word OBLIGE, without letting the letters B,E be together? so, I take all the letters except B,E, putting one space between them where I can insert the two letters I said before. See the following schema: _ O _ L _ I _ G _ I) here, each time I select one letter, then I go to select one space that will host this last letter, so: I go to choose 1 out of 2 letters {B,E} and also 1 out of 5 spaces for the letter just selected, and, I go to choose 1 out of 1 remaining letter and 1 out of 4 remaining spaces for the letter just selected: $\displaystyle \binom{2}{1} \cdot \binom{5}{1} \cdot \binom{1}{1} \cdot \binom{4}{1} = (2 \cdot 5) \cdot (1 \cdot 4) = 10 \cdot 4 = 40$ but, this result is not correct. the right reasoning would be the following: II) At first, I consider every necessary space to insert the letters I want, and then I go to select the wanted letters: $\displaystyle \binom{2}{1} \cdot \binom{1}{1} \cdot \binom{5}{2} = 2 \cdot 1 \cdot 10 = 20$ identical result of: $\displaystyle {}^5P_{2} = 5 \cdot 4 = 20$ the letters OLIG can be arranged in 4!, so, the number of ways to arrange the entire word is: $\displaystyle 4! \cdot 20 = 480$ my question is: even if both reasonings seems to me logically right, please, can you explain me, why it's wrong the reasoning I), but it's right the reasoning II) ? many thanks! March 9th, 2015, 12:24 PM   #2
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Quote:
 In how many ways can you arrange the letters of the word OBLIGE, without letting the letters B,E be together?

_ O _ L _ I _ G _ Tags combinatorics, letters, select, spaces Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post beesee Algebra 7 February 14th, 2015 10:53 AM beesee Algebra 10 February 3rd, 2015 12:48 AM Kimi Math 1 December 1st, 2014 04:17 PM Solarmew Applied Math 2 April 20th, 2012 12:11 PM mad Algebra 10 August 28th, 2010 09:32 PM

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