Discrete Math 2-Tutorial 7- Combinations w/repetition Part 2

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Completing the example from the last video.
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Thanks for these videos! I just have one question: at 2:50 Don't the Xi's start out being greater than zero, and the Yi's become greater than or equal to zero? If either X1 or X3 is equal to zero, then it means you are possibly getting no Heads occurring during the first or third run of Heads  right? But in order to have a run you need at least one Heads to occur. Also, if Yi > 0, and Xi = Yi + 1, then Xi must be greater than zero also right? 

droitnow
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why are we subtracting 1 from the x variables?

onePunch