M26-13

Official Solution:

Anna has 10 marbles: 5 identical red, 2 identical blue, 2 identical green, and 1 yellow. She wants to arrange all of them in a row such that no two adjacent marbles are of the same color, and the first and last marbles are of different colors. How many different arrangements are possible?

A. 30
B. 60
C. 120
D. 240
E. 480


Seems tough and complicated, but if we read the stem carefully, we find that the only way both conditions can be met for the 5 red marbles, which are half of the total marbles, is that they can be arranged in only two ways: R*R*R*R*R* or *R*R*R*R*R.

Here comes the next good news: in these cases, BOTH conditions are met for all other marbles as well. No two adjacent marbles will be of the same color, and the first and the last marbles will be of different colors.

Now, it's easy: 2 blue, 2 green, and 1 yellow marble can be arranged in 5 empty slots in \(\frac{5!}{2!*2!}=30\) ways (this is the permutation of 5 letters BBGGY, out of which 2 B's and 2 G's are identical). Finally, as there are two cases (R*R*R*R*R* and *R*R*R*R*R), the total number of arrangements is \(30*2=60\).


Answer: B