I was preparing for my Engineering entrance exam and got across this question.

A flag is to be made with 6 vertical stripes by using colours yellow, blue, green and red in such a way that none of the adjacent stripes have same colour. How many ways is it possible?

I landed up with 12x81 but I tried again and landed with 24x216

  • $\begingroup$ if you explain how you counted we can find possible errors $\endgroup$ – mm-aops Jun 21 '14 at 18:27
  • $\begingroup$ @mm-aops first I took yellow in position 1 then 2 then on till 6 later green, red, blue. So 4 colours and 6 different positions. 6x4=24. There was an option given there 24x216 so i thought it might be right. $\endgroup$ – user3627194 Jun 21 '14 at 18:32
  • $\begingroup$ Just worked on this same type problem over here: math.stackexchange.com/questions/840965/permutation-combination/… $\endgroup$ – bobbym Jun 21 '14 at 18:40
  • $\begingroup$ @bobbym ok. But how did u expand ( 1111), that vertical one in end $\endgroup$ – user3627194 Jun 21 '14 at 18:42
  • $\begingroup$ Hi; I had to use Mathematica, too tedious to do by hand. You will find more manageable solutions there too. $\endgroup$ – bobbym Jun 21 '14 at 18:44

what you do is you consider the first strip - its colour can be chosen out of all possible $4$. once you chose the colour of the first one you only have $3$ choices for the second one - cause it can't be the same which gives you $4 \cdot 3$ possibilities for the first two stripes. continuing this way you end up with $4 \cdot 3^5$ ways

  • $\begingroup$ There's no option as such there. $\endgroup$ – user3627194 Jun 21 '14 at 18:36
  • 1
    $\begingroup$ I'm pretty sure that's correct unless you have extra assumptions in your problem - i.e. all the colors have to be used $\endgroup$ – mm-aops Jun 21 '14 at 18:41
  • $\begingroup$ I agree that is correct. It also is one of his options of 12 x 81. $\endgroup$ – bobbym Jun 21 '14 at 19:09

It must be 12 x 81 for sure. It is not that hard. For six strips the first can be filled up in 4 ways, i.e, any one of the colours ; and the next can be filled in 3 ways since the colour used in the first strip cannot be used. The third can be filled up again in 3 ways since the colour to the left cannot be used. And this goes on till the last strip. So we finally get 4 x 3^5. Equivalent to 12 x 81.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.