The vertices of a regular pentagon are to be colored using five differ

Question

The vertices of a regular pentagon are to be colored using five different colors. In how many ways can the pentagon’s vertices be colored if the 5 colors are to be chosen from a palette of 6 different colors?

A. 64 
B. 96 
C. 108 
D. 144 
E. 192

rxb266 · Accepted Answer

Solution:
 we can choose 5 colours from a total of 6 colours in 6C5 ways ie 6 ways !!
we can arrange these 5 chose colours in (5-1!) ways ( since the arrangement is similar to the circular arrangement)

thus total no. of ways: 6C5 x (5-1)! = 144 ways

nick1816 · Answer

Number of ways to select 5 different colors out of 6 different colors= 6C5=6

Number of ways to arrange 5 different colors on the vertices of regular pentagon= 5!/5=24

Number of ways the pentagon’s vertices can be colored if the 5 colors are to be chosen from a palette of 6 different colors= 6*24= 144

Kinshook · Answer

Total no of ways pentagon's vertices be colored = 6C5 * 4! = 6*24 = 144

IMO D

philipssonicare · Answer

Combinations/permutations type questions, unless extremely basic, are much harder than sub-600. No chance this is sub-600.

MathRevolution · Answer

=>

The number of ways to choose  5  colors out of  6  colors is  6 C 5  =  6 .

The number of circular permutations of the  5  colors is  (5-1)! = 4! = 24.

Thus,  6*24 = 144  different colorings of the pentagon are possible. 
 
Therefore, D is the answer.
Answer: D

prabsahi · Answer

Hello Mathsrevolution,

Can you please explain the highlighted part

The number of circular permutations of the  5  colors is  (5-1)! = 4! = 24.

MathRevolution · Answer

The number of ways to arrange n distinct objects along a fixed (i.e., cannot be picked up out of the plane and turned over) circle is (n-1)!. 
The number is (n-1)! instead of the usual factorial n! since all circular permutations of objects are equivalent because the circle can be rotated.

kheraarj · Answer

How do we know that the order of color doesnot matter in this case?

Suppose that we have green, red, blue, yellow, brown, black-- 6 colors

We can have blue first, then an option of any of the 5 colors in the second wall, then any 1 of the 4 in the 3rd, then any of 3 in the 4th, and then any of the 2 in the 5th.

There are 6 ways to do the same (for each color), therefore answer should be 6!

Why is the above wrong?

Posted from my mobile device

Stanindaw · Answer

Will we use (n-1)! for triangle too??

bumpbot · Answer

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The vertices of a regular pentagon are to be colored using five differ

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The vertices of a regular pentagon are to be colored using five differ

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