2014 AMC 10A Problem 4

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 permutations casework

Difficulty rating: 1140

Small hint:

Case on where the yellow house appears

Big hint:

Use the blue-before-yellow condition and the no-adjacent condition together

Solution:

Blue must come before yellow but not next to it, so they sit in positions

(1,3)

or

(1,4)

or

(2,4)

.

In each case orange and red fill the two remaining spots with orange before red, which is forced. The three orderings are orange, blue, red, yellow; blue, orange, red, yellow; and blue, orange, yellow, red.

There are

3

possible orderings.

Thus, B is the correct answer.