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 Catering to sportsmen's whims (Posted on 2011-09-04)
Adam, Bob and Charles, members of a certain team, insisted on sticking together, while taking their places on a bench, accommodating the whole team of n people, - while Dan and Eddy would not like to be placed one next to another.
In how many ways may all these requests be met, if n = 5, 8, 10.

Rem: Computer programs allowed only as verification of your analytically produced results.

 No Solution Yet Submitted by Ady TZIDON Rating: 5.0000 (1 votes)

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 solution | Comment 2 of 4 |

A, B and C initially count as one person in placing the teammates, but then the solution count has to be multiplied by 6 to account for the shuffling within that group.

Initially that gives us 6 * (n-2)!, but from this we must subtract out those orders in which D and E are together. That can be calculated in the same manner as A, B and C together: twice what you'd get considering D and E as one person. So that comes to 12 * (n-3)!

The answer is then 6 * (n-2)! - 12 * (n-3)!.

` N     5    36 - 24 = 12 8    4320 - 1440 = 288010    241920 - 60480 = 181440`

Extending, with computer program:

`list   10   for N=5 to 20   20    Ways=6*!(N-2)-12*!(N-3)   30    print N,Ways   40   nextOKrun 5       12 6       72 7       432 8       2880 9       21600 10      181440 11      1693440 12      17418240 13      195955200 14      2395008000 15      31614105600 16      448345497600 17      6799906713600 18      109844646912000 19      1883051089920000 20      34145993097216000`

 Posted by Charlie on 2011-09-04 13:28:34

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