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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 combinations.Combinations 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 combinations is the order in which the items are picked does not make a difference. We are only looking for a group of items and it does not matter the order in which they get in the group. For example, suppose we have a club and we want to pick two people to work on a committee. Suppose the first person picked is Jane and the second one picked is Joe. Both are on the committee and it did not matter who was picked first. The order in which the two were picked did not make a difference in the outcome of forming the committee.The formula we are going to use to calculate combinations, denoted by nCr, 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 r!(n-r)!. Finally, we divide the two. Now we will look at an example.
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