Problem:
A child builds towers using identically shaped cubes of different colors. How many different towers with a height of cubes can the child build with rod cubes, blue cubes, and green cubes? (One cube will be left out.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Of the cubes available, cube will not be used. Because there are three different kinds of cubes and limited numbers of each kind, there are three different possibilities for the set of cubes that are used. One possibility is red cube, blue cubes, and green cubes; the second possibility is red cubes, blue cubes, and green cubes; and the third possibility is red cubes, blue cubes, and green cubes. Cubes of the same color are indistinguishable. Hence the number of different towers is
There is a one-to-one correspondence between towers of height and towers of height by viewing the top cube in a tower of height as the cube that is not used in a tower of height . The number of different towers of height is given by
Note: A generalization of the two solutions put together yields the following extension of Pascal's Identity:
The problems on this page are the property of the MAA's American Mathematics Competitions