There are 2n+1 identical books to be put in a bookcase with three shelves. In how many ways can this be done if each pair of shelves together contains more books than the other shelf?
x1 = number of books on shelf 1, x2 = number of books on shelf 2, x3 = number of books on shelf 3
I have the following inequalities:
a+b > c
a+b+c > 2c (don't understand where this one comes from)
2n+1 > 2c
c < n+ (1/2)
c is an integer, so c =< n. Same for a, and b.
=> a+b+c = 2n+1, where 0 =< a, b, c =< n
At this point, I don't know how to finish the computation of the number of integral solutions. The answer is (n+1) choose (2).
I would have thought that r = 2n+1, and k = 3, so the answer would have been: (2n+3) choose (2).