Permutations Essay

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Permutations There are basically two types of permutation: 1. Repetition is Allowed: such as the lock above. It could be "333". 2. No Repetition: for example the first three people in a running race. You can't be first and second. 1. Permutations with Repetition These are the easiest to calculate. When you have n things to choose from ... you have n choices each time! When choosing r of them, the permutations are: n × n × ... (r times) (In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.) Which is easier to write down using an exponent of r: n × n × ... (r times) = nr Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them: 10 × 10 × ... (3 times) = 103 = 1,000 permutations So, the formula is simply: nr | where n is the number of things to choose from, and you choose r of them (Repetition allowed, order matters) | 2. Permutations without Repetition In this case, you have to reduce the number of available choices each time. | For example, what order could 16 pool balls be in?After choosing, say, number "14" you can't choose it again. | So, your first choice would have 16 possibilites, and your next choice would then have 15 possibilities, then 14, 13, etc. And the total permutations would be: 16 × 15 × 14 × 13 × ... = 20,922,789,888,000 But maybe you don't want to choose them all, just 3 of them, so that would be only: 16 × 15 × 14 = 3,360 In other words, there are 3,360 different ways that 3 pool balls could be selected out of 16 balls. But how do we write that mathematically? Answer: we use the "factorial function" | The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples: * 4! = 4 × 3 × 2 × 1 = 24 * 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 * 1! = 1 | Note:

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