# In how many ways can we form a garland using distinct red, yellow, and blue flowers if flowers of the same colour must be together?

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.

• 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. – N. F. Taussig May 5 '18 at 1:12
• I did 1/2 (n-1)! For finding arrangements of 3 groups – Niki May 5 '18 at 1:18
• That gives you one arrangement, which makes sense if you are equating reflections. If not, you should have two arrangements of the three colours. – N. F. Taussig May 5 '18 at 1:19

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.