Counting the number of outcomes in games of chance has been a popular pastime for many centuries. This was of interest not only because of the gambling that was involved, but also because the outcomes of games of chance were often interpreted as divine intent. Thus, it was just about a thousand years ago that a bishop in what is now Belgium determined that there are 56 different ways in which three dice can fall provided one is interested only in the overall result and not in which die does what. He assigned a virtue to each of these possibilities and each sinner had to concentrate for some time on the virtue which corresponded to his cast of the dice. (a) Find the number of ways in which three dice can all come up with the same number of points.
(b) Find the number of ways in which two of the three dice can come up with the same number of points, while the third comes up with a different number of points.
(c) Find the number of ways in which all three of the dice can come up with a different number of points.
(d) Use the results of parts (a), (b), and (c) to verify the bishop's calculations that there are altogether 56 possibilities.