How many? (3)

Let us suppose that the England football manager (or coach, as we now apparently have to call him) has chosen a squad of 15 players for a certain game, and now must pick his team from this squad. Assuming any player can play in any position, how many different teams can he come up with? This is somewhat different from what we have discussed before, because here order doesn't matter. All that matters is which 11 players are chosen - the order in which they are chosen is of no consequence. So how do we calculate this? Well, if order did matter, we know from last time that the answer would be the first 11 terms of factorial 15, that is 15x14x13x12x11x10x9x8x7x6x5 = 54,486,432,000. But let's just look at one possible team - players A,B,C,D,E,F,G,H,I,J and K. These 11 players as we know could be arranged in 11! different ways, but for the purpose of choosing a team, this is just a single grouping. The same goes for any other 11 players, so what we have to do is divide 54,486,432.000 by 11! (39,916,800) and this comes to 1,365. A non-ordered grouping is called a combination, and the formula is that the number of combinations of N things selected from T things is T!/N!(T-N)!. Here T is 15, N is 11, so T-N is 4, and that gives us 15x14x13x12x11x10x9x8x7x6x5x4x3x2x1/11x10x9x8x7x6x5x4x3x2x1x4x3x2x1 and cancelling out top and bottom leaves us with 15x14x13x12/4x3x2x1 which comes to 1,365.
We'll draw all this together next time.