A track and field championship has participants from 64 countries. The flag of each country consists of three horizontal stripes of different colors where order matters (every flag has a fixed top, middle, and bottom stripe color). No flag contains colors other than red, white, blue, and green. Show that there are 3 countries that have the same flag.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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A track and field championship has participants from 64 countries. The flag of each country consists of three horizontal stripes of different colors where order matters (every flag has a fixed top, middle, and bottom stripe color). No flag contains colors other than red, white, blue, and green. Show that there are 3 countries that have the same flag.

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Step 1

 Details of the flag of countries participating : 

  1. The flag of each country consists of 3 horizontal stripes .
  2. The stripes are of different colors where order matters .
  3. no flag contains colors other than red, white, blue, and green.

First , we consider the number of possible ways a country can make a flag fulfilling all the above conditions

mentioned in points 1. , 2.  and 3.

As a flag contains 3 horizontal stripes , we have to decide what color each stripe would be .

  • We start with the top most stripe of the flag.

As per condition 3,No flag contains colors other than red, white, blue, and green

Therefore , We have only 4 choices for colors Red ,White , Blue , Green , for the top most stripe .

  • Now , we consider the middle stripe of the flag 

Also ,according to condition 2  each stripe should be of  different color, so we cannot repeat a color in one flag.

that is ,for instance ,  if we choose the top most stripe will be of red color then middle stripe of the flag cannot be red .So , we are left with 3 choices ( white , blue , green ) for the middle stripe of the flag.

  • Next , we consider the bottom stripe of the flag.

Again , by condition 2 , this stripe should have a color different from the other two ( top stripe and middle ) 

So , after choosing a color for  the top and middle stripes of the flag , we are left with 2 choices for the bottom stripe of the flag.

Therefore from the above discussion we can conclude that the number of possible ways a country can make a flag fulfilling all the given conditions = 42 = 24 .

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