1989 AMC 12 Problem 22

Question

A child has a set of

96

distinct blocks. Each block is one of

2

materials (plastic, wood),

3

sizes (small, medium, large),

4

colors (blue, green, red, yellow), and

4

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.)

29

39

48

56

62

Answer: A

Concepts:casework combinations multiplication principle

Difficulty rating: 1600

Small hint:

For each attribute, count the alternatives different from the specified block

Big hint:

Choose a pair of attributes to change, then multiply their alternative counts

Solution:

The numbers of alternatives for material, size, color, and shape are

1,2,3,3.

Changing exactly two attributes gives

\begin{aligned} &1\cdot2+1\cdot3+1\cdot3\\ &\qquad+2\cdot3+2\cdot3+3\cdot3=29. \end{aligned}

Thus the correct answer is A.