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How do you know their are 21 total possible outcomes? I could draw out the picture in a graph and count them but is their a more intuitive way of being able to determine this?

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    A recommendation to the poser: Title your question with key words relevant to the true problem at hand. Your question has nothing whatsoever to do with "100 shares" and "200 shares" for instance. This will help future questioners and answerers find relevant prior questions and answers. – 2017-02-19
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    Thanks for the recommendation @David G. Stork. I apologize, I am new to the site so the conventions are unfamiliar to me. – 2017-02-20

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For general linear constraints determining the cardinality of an integer feasible region can be hard. In this case though, simply note that $x$ can take $7$ distinct values and that for each of those values $y \in \{x-1,x,x+1\}$. Thus there are $7\cdot 3=21$ feasible points.

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    Thank you for answering my question. This is exactly the intuition I was looking for. I will keep in mind that for general linear constraints the cardinality of such regions can be hard to determine. I just wanted to see how to attack this problem in a cleaner and quicker way. Thanks! – 2017-02-20
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    @allan not a problem, glad it was helpful! – 2017-02-20