## Friday, April 6, 2012

### JCCDQBHWHCB007(ii) Combinatorics : “Choose, then Automatically fill” technique

The question says “the captain must stand between the two youngest players” which I interpret literally to mean: not necessarily directly next to each other, but there can be intervening players.  Although I suspect that the one who set this question might have meant that the captain is supposed to be in between directly next to the two youngest players, I shall attempt the question according to its prima facie meaning.  Later, I shall consider what if the captain and two youngest players are together next to one another with the captain in the middle.

Suggested Approach and Solution:-

Number of rows to choose from = 2

The condition that “the captain must stand between the two youngest players” seems difficult, because there can be one or more intervening players.  Generally we do not favour complicated case-by-case analyses.  The good news is that we can use a “choose, then automatically fill” technique to tackle this.  Within the row with the captain, there are 5 positions and we choose 3 of them.  Then we automatically fill them with a youngest player, the captain and then the other youngest player.  The number of ways to do this is  5C3 = 10.

Number of ways the two youngest players can swap around = 2!.
Number of ways the rest of the players can shuffle around   = 7!.

Total number of ways = 2 x 5C3 x 2! x 7! = 201 600.

What if …
What if the captain is in between directly next to the two youngest players?

Number of rows to choose from = 2

If the captain is in between and directly beside the two youngest players, we treat these 3 players as a unit.  There are 3C1 = 3 ways to position this group within the row.  Note that within this group, the two youngest players can still swap around while the captain stays in their middle.  Everything else is the same as above.

Total number of ways = 2 x 3C1 x 2! x 7! = 60 480.