Problem:
A child has a set of distinct blocks. Each block is one of materials (plastic, wood), sizes (small, medium, large), colors (blue, green, red, yellow), and shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the "plastic medium red circle" in exactly two ways? (The "wood medium red square" is such a block.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are ways a block can differ from the given block in exactly two ways: material and size, material and color, material and shape, size and color, size and shape, and color and shape. Since there is only choice for a different material, choices for a different size, choices for a different color, and choices for a different shape, it follows that the number of blocks in each of the above categories is , and , respectively. The answer is the sum of these six numbers.
Note. The number of blocks that differ from the given block in exactly ways is the coefficient of in . This is an example of a generating function.