AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 2 Medium

Given 6 different colors of paint to color the six regions A,B,C,D,E, and F in the figure, with the condition that adjacent regions cannot be painted the same color, how many different colorings are there?

  • A.

    3880

  • B.

    3180

  • C.

    6120

  • D.

    3240

  • E.

    5180

Answer:C

Assuming the coloring order is D-C-E-F-A-B,

 

If C and E are colored the same, then there are

6\times 5\times 1\times 4\times 4\times 3=1440 ways.

 

If C and E are different, and A and E are the same, then there are

6\times 5\times 4\times 3\times 1\times 4=1440 ways.

 

If C and E are different, and A and E are also different, then there are

6\times 5\times 4\times 3\times 3\times 3=3240 ways.

 

Therefore, by the Addition Principle of Counting, the total number of colorings is

1440+1440+3240=6120.

 

Hence the answer is 6120.

Related Problems
Problem 1 AMC 10 Weekly Practice Round 3
Problem 3 AMC 10 Weekly Practice Round 3
Problem 4 AMC 10 Weekly Practice Round 3
Problem 5 AMC 10 Weekly Practice Round 3
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