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Sep 10th, 2015

Explanation:

Case 1: The case where she makes 2 sales(S)

She cannot get any ‘fails’ (F),or the log will end immediately.

She can also get from 0 to 4 tentatives (T). (she does not have to get the full tentatives because she made 2 sales)

Let n(S), n(T), n(F) denote the number of sales, … etc.

So, when the log ends with n(S) = 2, n(F) = 0 and n(T) = 0, 1, …, 4

With n(T) = 0, there is just 1 possible log: S, S.

With n(T) = 1, there are 2 possible logs: T, S, S and S, T, S.

With n(T) = 2, the number of logs is the number of ways of choosing which position 1, 2 or 3 the S will come,=3

With n(T) = 3, the number of logs is 4.

With n(T) = 4, the number of logs is 5.

Add these totals to find the number of possible logs for case 1
Total for n(S) = 2 is 1 + 2 + 3 + 4 + 5 = 15

Thus case 1=15

Case 2
If n(T) = 5, then once again n(F) = 0, and n(S) = 0 or 1.

If n(S) = 0, there is just 1 log.

If n(S) = 1, there are 5 positions in which the S could come.

Add these totals to find the number of cases for case 2
So the total for n(T) = 5 is 1 + 5 = 6

Case 3
If n(F) = 1, then the F must come at the end of the sequence, with n(S) = 0, 1 and n(T) = 0, 1, …, 4

If n(S) = 0, there are 5 possible logs, corresponding to each of the five values of n(T).

If n(S) = 1, there are 1 + 2 + 3 + 4 + 5 = 15 possible logs (each of these individual numbers being the number of ways of positioning the S with 0, 1, …, 4 T’s)

Add these totals to find the number of case 3
So the total for n(F) = 1 is 5 + 15 = 20

Add all the totals from case 1, case 2, and case 3
Case 1+Case 2+Case 3= 15 + 6 + 20 = 41 logs.

Edited some of it so it made a little more sense, but the math is sound.

Please let me know if you need any clarification. I'm always happy to answer your questions. Leave a positive review if you are happy!
Sep 10th, 2015

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Sep 10th, 2015
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Sep 10th, 2015
Oct 20th, 2017
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