Why does the first set in exercise 3.9 in which the tallest bookcase is red have the answer as 1x3x4^4 and not 1x4^4? Why does the second bookcase only have three options as opposed to four?
paulinalin: Jan. 15, 2015, 11:42 p.m.
I think the 3 refers to the shortest three bookcases that have to be the same color.
weisbart: Jan. 17, 2015, 10:32 a.m.
That's right Paulina! The point is that we use a disjoint partition: The tallest bookcase is red or not red. The one comes from the fact that the tallest is red, there is but one choice. The shortest three are painted the same color and there are three colors, hence the three. We then have four remaining bookcases which may be painted one of three colors each or left unpainted, this gives $4^4$ possibilities. \[\] Note that in the case when the tallest is not red, we have three things that could be done to it: it can be left unpainted or it could be painted one of two colors (not red). In this case, anything can be done to the seven remaining bookcases, hence $4^7$ possibilities.
martha2A: Jan. 18, 2015, 11:07 p.m.
Oh now I see! My problem was that I thought that whatever color the tallest bookcase was painted then the three shortest bookcases would also be painted that color but I was wrong the only restriction in this problem is if the tallest bookcase is painted red. (posted my mistake in thought process just in case anyone else was thinking the same thing)