How many? (2)

We saw last time that five people (or things for that matter) can be arranged in 5x4x3x2x1 = 120 different ways. Following the same logic, myself, my children and grandchildren number 10 in all, and so the number of different ways in which we can be arranged is the almost unbelievable 10x9x8x7x6x5x4x3x2x1 = 3,628,800. This procedure of multiplying a number by all the numbers less than it down to 1 is called a factorial, and is written by the number followed by an exclamation mark. Arranging things in different order in this way is called permutation, so we can say that the number of permutations of 5 things is 5! (read as "factorial 5"), that of 10 things is 10!, or in more general terms that the number of permutations of N things is N!. Now there used to be a game in a daily newspaper where you were given seven letters and invited to make as many three-letter words out of these letters that you could. One way of approaching this would be the "brute force" approach of making all the possible different 3-letter arrangements and then seeing which of them are allowable words. But just how many such arrangements are there? This in fact is just a simple variation on what we've already learned. We have three imaginary pegs where the letters can go, and any of the seven can be in position 1, any of the remaining 6 in position 2 and any of the remaining 5 in position 3. So the answer is 7x6x5 = 210. In strict mathematical terms we say that the number of permutations of N things selected from a total of T things is T!/(T-N)!. Here T is 7, N is 3, so T-N is 4, so we have 7!/4! or 7x6x5x4x3x2x1/4x3x2x1. and the 4x3x2x1 cancels out top and bottom which leaves us with 7x6x5.

More to come.