Flag making with 6 vertical stripes

Question

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

@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. — user3627194, Jun 21 '14 at 18:32
Just worked on this same type problem over here:
http://math.stackexchange.com/questions/840965/permutation-combination/842434#842434 — bobbym, Jun 21 '14 at 18:40
@bobbym ok. But how did u expand ( 1111), that vertical one in end — user3627194, Jun 21 '14 at 18:42
Hi; I had to use Mathematica, too tedious to do by hand. You will find more manageable solutions there too. — bobbym, Jun 21 '14 at 18:44
@bobbym some or all colors. Options here are 12x18, 24x216, 16x108. I don't know which is correct. Also if you know what type of sums are these then please tell me. I would like to study them in deep. — user3627194, Jun 21 '14 at 18:48

score 6 · Accepted Answer · answered Jun 21 '14 at 18:34

6

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

answered Jun 21 '14 at 18:34

mm-aops

4,195

There's no option as such there. – user3627194 Jun 21 '14 at 18:36
1

I'm pretty sure that's correct unless you have extra assumptions in your problem - i.e. all the colors have to be used – mm-aops Jun 21 '14 at 18:41
I agree that is correct. It also is one of his options of 12 x 81. – bobbym Jun 21 '14 at 19:09

score 0 · Answer 2 · answered Mar 15 '17 at 18:38

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.

Flag making with 6 vertical stripes

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