document.write( "Question 1128792: Let A and Between sit around table with twelve other person. What is probability that exactly three person between them? \n" ); document.write( "

Algebra.Com's Answer #745324 by ikleyn(52834) $\"\"$ $\"About$

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\n" ); document.write( "Let A and $\"highlight%28cross%28Between%29%29\"$ B sit around table with twelve other person. What is probability that $\"highlight%28there%29\"$ $\"highlight%28are%29\"$ exactly three person between them?
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\n" ); document.write( "\n" ); document.write( "            There are two ways to solve this problem.\r
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\n" ); document.write( "\n" ); document.write( "            One way is \"hard\": it is to count the number of circular permutations.\r
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\n" ); document.write( "\n" ); document.write( "            The other way is \"easy\": it is based on \"common sense\".\r
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\n" ); document.write( "\n" ); document.write( "            I prefer the easy way, so I'll present it first.\r
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\n" ); document.write( "\n" ); document.write( "Easy way solution\r
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document.write( "This problem is about \"circular permutations\", so we can assume that all the chairs around the table are numbered sequentially \r\n" );
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document.write( "from 1 to 2+12 = 14  inclusively and that the person A is sitting on the chair #1.\r\n" );
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document.write( "Then person B can occupy any of the rest 13 chairs, but the chairs #5 and #11 are favorable.\r\n" );
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document.write( "So, the probability under the question is the ratio  .     ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "\"Hard way\" solution\r
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document.write( "This problem is about \"circular permutations\", so we can assume that all the chairs around the table are numbered sequentially \r\n" );
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document.write( "from 1 to 2+12 = 14  inclusively and that the person A is sitting on the chair #1.\r\n" );
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document.write( "Then the person B is sitting EITHER on the chair #5  OR  on the chair #11.\r\n" );
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document.write( "    In the first case, we have 12*11*10 ways to place first 3 of remaining 12 persons on the chairs #2, #3 and #4,  \r\n" );
document.write( "    and we have  9*8*7*6*5*4*3*2*1 ways to place the rest of 12 people on the remaining chairs.\r\n" );
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document.write( "    It gives us  12! different ways.\r\n" );
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document.write( "    In the second case we have THE SAME NUMBER of different ways.\r\n" );
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document.write( "    Hence, in all, there are 2*12! different placements/arrangements that satisfy the condition (favorable arrangements).\r\n" );
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document.write( "    The total number of circular arrangements of 14 persons is 13!, as it is well known.\r\n" );
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document.write( "    So, the probability under the question is the ratio  , which is equal to  .\r\n" );
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document.write( "    We got the same answer.\r\n" );
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\n" ); document.write( "\n" ); document.write( "See the lesson\r
\n" ); document.write( "\n" ); document.write( " - Persons sitting around a cicular table \r
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