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In how many ways can we form a garland using $3$ different red, $5$ different yellow and $4$ different blue flowers, if flowers of the same colour must be together?

My approach:

I made $3$ groups of $3$ different flower colours and applied circular permutations clockwise, then I found internal permutations and multiplied, but this approach isn’t giving me the correct answer.

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  • $\begingroup$ If you start with red, there are only two ways to arrange the colour groups, rby and ryb, as you proceed clockwise around the circle. $\endgroup$ – N. F. Taussig May 5 '18 at 1:12
  • $\begingroup$ I did 1/2 (n-1)! For finding arrangements of 3 groups $\endgroup$ – Niki May 5 '18 at 1:18
  • $\begingroup$ That gives you one arrangement, which makes sense if you are equating reflections. If not, you should have two arrangements of the three colours. $\endgroup$ – N. F. Taussig May 5 '18 at 1:19
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First, arrange the colours. If we proceed clockwise from red, there are two ways to do so, $rby$ and $ryb$. Now, arrange the groups of flowers of each colour internally.

For invariance under rotation, we obtain $$2 \cdot 3!5!4!$$ For invariance under rotations and reflection, we divide this result by $2$ to equate clockwise and anti-clockwise rotations.

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  • $\begingroup$ Thanks I will try this way answer to this problem is 17280 ways $\endgroup$ – Niki May 5 '18 at 1:26
  • $\begingroup$ In that case, we must account for invariance under reflection. $\endgroup$ – N. F. Taussig May 5 '18 at 1:29

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