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Often in probability we wish to count the number of ways something can be done such as chosing a number of items from a set. This is frequently difficult to do manually as the numbers of ways can be extremely large. Because there can be various conditions on how the items are chosen, there are various techniques available to count the many ways. One such method is called permutations.
Permutations are used when we are picking items from a set and the items are not put back so they can be picked again. This is know as choosing without replacement. So the pool of items is being reduced each time a choice is made. But another critical feature that must be present when we use permuations is the order in which the items are picked makes a difference. There is some special significance for the first one picked, another significance for the second one picked, and so forth. For example, suppose we have a club and we want to pick a president and vice president. Suppose also that the first person picked, say Jane, becomes president while the second one picked, say Joe, becomes vice president. Jane is president because she was picked first. Now suppose Joe was picked first and Jane was picked second. Joe now is president and Jane is vice president. The order in which the two were picked makes a difference in the outcome.
The formula we are going to use to calculate permutations, denoted by _{n}P_{r}, is
where n is the number of items availble for selection and r is the number chosen. Incidentally, n! means to multiply together every integer from 1 to n. The same applies to r!. So the first step is to calculate the numerator n!. The second step is to calculate the denominator (nr)!. Finally, we divide the two. Now we will look at an example.
Find _{7}P_{7} 1. Find n!  7! = 5040  2. Find (nr)!  ( 7  7 )! = 0! = 1  3. Divide the numerator by the denominator  5040 / 1 = 5040 
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