A Boat Race

        The yacht club recently held a boat race. The captain had some difficulty deciding the order of cowing in the boat for his crew.

        3 of his crew were stroke side oarsmen only and 2 of them were bow side oarsmen only. Weights and personal preferences were not important really.

        In how many ways the captain could have arranged his eight men to form the crew?

Answer:  1728: The Captain is bound to go crazy deciding the order of rowing in the boat with so many possibilities on hand.

               Whichever way you work out the problem you obtain this number 1728. 

                First of all let us consider the stroke side men first. The fourth oarsman can be chosen from the 3 who can row on either side in 3 ways. And so when this fourth oarsman is chosen the four stroke side oarsman can be arranged in 4! Ways. Therefore there are 3x4! Ways of arranging the stroke side.

                Now let us consider the bow side oarsmen. There is no choice of men here, because there are 2 bow side oarsmen and the 2 can row on either side. These two men can be arrangement any of the bow side arrangements is possible.

 Thus the total number of arrangements is

3 x 4! x 4!  = 72x 24 = 1728