# Entrance exams

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The vertices A,B,C,D of a square are to be coloured with one of the 3 colours red,blue or green such that adjacent vertices get different colours.What is the number of such colourings?

a)18  b)12  c)20  d)24

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Imagine a Square ABCD. We have to color the vertices of the square such that adjacent vertices get different color.

We have three colors namely red,blue and green to complete the above task.Let us consider the point A, which can be colored by any of the three available colors. Hence total number of ways in which A can be colored is 3.

Now to color B such that it has a different color from A, which can be done by the remaining of the two colors in 2 ways.

Also we have 2 choices of color for point C so as that it has a different color from B, therefore C can be colored in 2 ways.

Finally, D is to be colored so that it has a different color from A and that of C, i.e., it has only one choice of the color, and can be colored in 1 way.

Therefore total number of ways to color the vertices A, B , C, D of the square is 3*2*2*1 ways= 12 ways…..b) is the correct option.