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Catering to sportsmen's whims (Posted on 2011-09-04) Difficulty: 3 of 5
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 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 = 2880
10    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   next
OK
run
 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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