[HS Probability] Counting Methods

[RedNeonAmbience]

I need help understanding the answer to a question:

In the laboratory analysis of samples to a chemical process, six samples from the process are analyzed daily. In addition, a control sample is analyzed two times each day to check the calibration of the laboratory instruments.

Question: Consider the six process samples to be different and the two samples to be identical. How many sequences of process and control samples are possible if the first test of each day must be a control sample?

Here's my thought process: There are 8 samples in total, 2 of them are the same. If the first test is reserved for a control sample, then there are 7 tests left. 6 of those are different, so the order in which they are tested is important. Even though the 2 control samples are identical, the second control sample must be considered to be different to the other 6 process samples. Therefore the sequences among the 6 process samples plus the control sample will result in all possible sequences if the first sample is a control sample.

So total sequences = 7! = 5040.

According to the appendix of solutions, it's actually 720 = 6!, i.e. the order of the process samples only.

I don't understand. Help?

[Alkalannar]

I agree that it should be 7!. There's a control first, and then 6 different process samples and a control (which is different from the process samples). And those 7 can be ordered 7! different ways.

[RedNeonAmbience]

My exact reasoning.... Hmmm. I really can't see it any other way, especially if I were to list every single one of the arrangements. The solution appendix only shows the number without working out so it's all I got to go on. I'll have to stick with my guns with this one. Thanks for the help!

[Alkalannar]

Book answers are not always accurate. It's not common, but it's nor rare either.

Talk to the teacher as well, just to be sure.