A country is divided into exactly 4 regions. Miki is sketching a map

Question

https://gmatclub.com/forum/download/file.php?id=73179 A country is divided into exactly 4 regions. Miki is sketching a map of the country's regions and plans to choose a color scheme for her map in which each region will be exactly 1 of 3 different colors and no adjacent regions will be the same color. One possible color scheme is for Regions 1,3 and 4 to have the same color and Region 2 to have a different color. How many color schemes are possible for Mikis map? A) 6 B) 8 C) 9 D) 24 E) 36 M1.png

Bunuel · Accepted Answer

Region 2 is adjacent to all three regions, while the remaining regions are adjacent only to Region 2 and thus depend only on Region 2's color.

Region 2 can be painted in one of three colors. Then, each of the remaining regions can be painted in any of the two remaining colors, making the number of different schemes equal to 3 * 2 * 2 * 2 = 24.

Answer: D.

gmatophobia · Answer

We know that Miki has three available colors, they can color the map using the following logic

Paint Region 1 in 3 ways ⇢ No restriction 
 Paint Region 2 in 2 ways ⇢ Any color except the one used in Region 1 
 Paint Region 3 in 2 ways ⇢ Any color except the one used in Region 2 
 Paint Region 4 in 2 ways ⇢ Any color except the one used in Region 2

Total = 3 * 2 * 2 * 2 = 24

Option D

Score-Quest · Answer

We have 3 colors and 4 regions. The condition is that no two adjacent regions can have the same color. Further, Regions 1, 3, and 4 are not adjacent to each other, but all are adjacent to Region 2.

So, we can calculate the number of possible color schemes as follows:
1. Choose a color for Region 2. There are 3 choices here
2. Choose colors for Regions 1, 3, and 4. Each region can be one of the two remaining (non-Region 1) colors with no other restriction since they are all non-adjacent regions. There are 2*2*2 choices here

Multiplying the choices for Region 2 with the combinations for Regions 1, 3, and 4, we get 3×8=24 possible color schemes.

So, there are 24 possible color schemes for Miki's map.

rey1009 · Answer

Clearly, 2 is adjacent to all and 1,3,4 are completely independent

so pick color of 2 and then assign colors to rest
3x{2 +3*2] = 3x8=24

curiousPope · Answer

gmatophobia  why are we not rearranging it in 4!/3! ways?

ashdank94 · Answer

Total way to arrange in 4 regions = 4!
Ways to arrange  with  the 2 being adjacent = 2!(4-2+1)!

Total color schemes: 4! - 2!(4-2+1)! = 12 x 2 ways = 24

sachi-in · Answer

This is a problem of graphs

1
|
|
2 ---- 4
|
|
3

2 is adjoint to 3 three but the rest 3 are not adjacent to each other

so if there are 3 choices for 2 then each regions will have 2 options to choose from

3 * 2 * 2 * 2 = 24 options

ShenotShih1017 · Answer

This is more like a logic puzzle than a math problem.

In general, I think it is best to approach these types of problems by picking a region and listing all the possible colors that could be assigned to it. In this case, region 1. If no other colors have been drawn on the map, region 1 can be colored any of the 3 colors.

If you move to region 2, since it borders region 1 and region 1 has already been assigned a color, region 2 can be either of the 2 colors not assigned to region 1.

Similarly, in region 3, as it borders region 2, it can be colored either of the 2 colors that region 2 is not.

And finally, as region 4 borders region 2, it can be either of the 2 colors that region 2 is not.

3x2x2x2 = 24.

SudsMeister · Answer

I tried to do it by counting cases. The main point to remember is that 2 can never have the same colour as 1, 3, or 4.

Colour 1 
 Colour 2 
 Colour 3 
 
  Case 1: 
 1, 3, 4 
 2 
 - 
 
  Case 2: 
 1, 3 
 2 
 4 
 
  Case 3: 
 1, 4 
 2 
 3 
 
  Case 4: 
 1 
 2 
 3, 4

Now, each case can arrange themselves in 3! ways i.e. 6 ways.

4 cases x 6 ways = 24.

Answer D.

Dream009 · Answer

mY approach was

read this as R1 R2 R3 R2 R4

_ 3 ways_ * 2 ways * 2 ways (3 - R2 colour) * R2 color already chosen * 2 ways
= 3*2*2* 2= 24 ways

A country is divided into exactly 4 regions. Miki is sketching a map

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

	Colour 1	Colour 2	Colour 3
Case 1:	1, 3, 4	2	-
Case 2:	1, 3	2	4
Case 3:	1, 4	2	3
Case 4:	1	2	3, 4

GMAT Problem Solving (PS) Questions

A country is divided into exactly 4 regions. Miki is sketching a map

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards