2014 AMC 12A Problem 3

Question

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

2

3

4

5

6

Answer: B

Concepts:arrangements with restrictions casework

Difficulty rating: 1200

Small hint:

Split into cases by whether the orange house or the blue house comes first

Big hint:

Because blue and yellow cannot be adjacent, at least one house must separate them

Solution:

If orange comes first, then blue and yellow cannot be adjacent, forcing the order orange, blue, red, yellow.

If blue comes first, yellow can be in the third or fourth position (never second, to avoid adjacency), giving blue, orange, yellow, red and blue, orange, red, yellow.

These are the only

3

valid orderings.

Thus, the correct answer is B.