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.

Prove that 6 * (n-2)! - 12 * (n-3)! = 6(n-3)!*(n-4) without merely listing the values of a finite set as listed below:

list
   10   for N=5 to 20
   20     Ways=6*!(N-2)-12*!(N-3)
   30     print N,Ways,6*!(n-3)*(n-4)
   40   next
OK
run
 5       12      12
 6       72      72
 7       432     432
 8       2880    2880
 9       21600   21600
 10      181440          181440
 11      1693440         1693440
 12      17418240        17418240
 13      195955200       195955200
 14      2395008000      2395008000
 15      31614105600     31614105600
 16      448345497600    448345497600
 17      6799906713600   6799906713600
 18      109844646912000         109844646912000
 19      1883051089920000        1883051089920000
 20      34145993097216000       34145993097216000
OK

Posted by Charlie on 2011-09-04 13:44:14