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.
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.
Details of the flag of countries participating :
- The flag of each country consists of 3 horizontal stripes .
- The stripes are of different colors where order matters .
- 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 = 4 x 3 x 2 = 24 .
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