Anna has 10 Marbles: 5 Red, 2 Blue, 2 Green, and 1 Yellow GMAT Problem Solving

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

1. 30
2. 60
3. 120
4. 240
5. 480

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

Seems difficult and challenging, but if we carefully study the stem, we discover that there are only two possible arrangements for the five red marbles—which make up half of the total marbles—in order to satisfy both conditions: R*R*R*R*R* or *R*R*R*R*R.

The next piece of good news is that in these situations, ALL other marbles satisfy BOTH requirements: neither the first nor the last marble will have the same hue as any nearby marble.
The solution is simple: 2 blue, 2 green, and 1 yellow can be put in 5 empty slots for a total of 5! / (2!2!) = 30 ways (permutation of the 5 letters BBGGY, where 2 B's and 2 G's are the same).
Last but not least, there are two examples (R*R*R*R*R* and *R*R*R*R*R), hence the total number of arrangements is 30
8^2 = 60.

B is the correct answer.

Approach Solution 2:

The fact that the marbles are identical is not revealed. Even if they might have the same hue, it's possible that they have distinct numbers or other differences that give each arrangement its own identity. In that case, the solution is different.

5!/2!*2! is because there are two blue and two green marbles.

5! = # of ways to set up five marbles
2! = repetition of two blue marbles
2! = repetition of green marbles.
This value is equal to 30.

A is the correct choice.

“Anna has 10 marbles: 5 red, 2 blue, 2 green, and 1 yellow" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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