How many ways can an IRS auditor select 3 of 8 tax returns for an audit?

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How many ways can an IRS auditor select 3 of 8 tax returns for an audit? Can you show me how you came up with the answer?

May 3rd, 2015

It depends on what you mean. If you care about the order the auditor selects the returns in, then there are 990 ways the auditor can select. If you only care about which returns get chosen, then there are 165. 

To explain, allow me to answer an easier question. Let's say the auditor is going to audit 1, 2, or 3 reports from the set {A, B, C}. 

How many ways can the auditor select 1 of these? 3 -- He can pick A, B, or C. 

Now, how many ways can the auditor select 2 of these? Let's say that the auditor has chosen A as the first report. Now, the auditor can pick either B or C, so there are 2 ways he can pick the second report. Therefore, for every first report, there are 2 ways the auditor can pick the second report. Therefore, there are 3*2 = 6 ways to select two reports. 

These 6 ways are {AB, AC, BA, BC, CA, CB}. Wait a minute -- isn't AC the same as CA? If you do not care about the order of selection (which I think you do not here), these should not be counted as different. How many times does each copy show up here? In this case, each pair shows up twice, so we need to divide by 2. Therefore, the number of ways that remain are 3*2/2 = 3, and these ways are {AB, AC, BC}. 

Technically, mathematicians call the first list the permutations of the set. The second list is called the combinations of the set. Note that permutations allow double counting, while combinations removes the double counting. 

You can see the wikipedia articles below for further reading, but here is how the other answers got different results. 

If you care about the order of the audit, then the number of ways is 

P(8,3) = 8 * 7 * 6 = 336

If you do not care about the order, you need to remove the copies. How many copies of each will you have? Well, how many ways can you arrange 3 different items? Well, you have 3 ways to pick the first one, 2 ways to pick the second one, and 1 way to pick the third one, so there are 3*2*1 = 6 ways to arrange three items. So, you must divide by 6. 

C(8,3) = 8*7*6 / (3*2) = 56 

Therefore, I am fairly confident that the answer to you question is 56, but more importantly I hope you understand how I came up with this number.

May 3rd, 2015

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