A little more Googling and I think the notation (8 choose 2) is how statisticians express combinatorials. So (8 choose 2) would be expressed in Excel as =COMBIN(8,2). That makes implementing the professor's solution fairly simple in Excel. But I don't understand the statistics enough to evaluate the professor's solution, nor do I understand enough to create the formula for a different problem.

If you are having trouble understanding the statistics, I would suggest getting the theory properly understood before worrying as much about how to implement the solution in Excel. At least one of the hits I got on Google was for a stat forum similar to this one. You might try posting this question somewhere like that to find people who understand the stat theory and can explain it to you. Then, if they can't help you implement it in Excel, come back here armed with an understanding of how to get the solution, and we can help figure out how to implement the solution in Excel. It would appear that implementation will primarily involve the COMBIN function.

Is there a way to have excel tell me what number turns up most within each row alongside 6?
Assuming the number selection is random and you go through enough trials or events, all numbers should have equal probability of occuring. If you are looking for a way to figure out what number occurs most in a limited number of trials, that should be doable. I'm not sure exactly how I would approach it, but it would probably consist of several COUNTIF's or SUMPRODUCT's which I could then add up to see which number occurs most frequently with a 6.