How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers? This is my first question here. 
I'm given $3$ white flowers and $6m$ red flowers, for some $m \in \mathbb{N}$. I want to make a circular garland using all of the flowers. Two garlands are considered the same if they can be rotated to be the same. My question is: how many different garlands can I make?
I tried an approach in which I divided the question into three parts:


*

*No. of ways in which the white flowers are all together

*No. of ways in which two white flowers are together

*No. of ways in which no two white flowers are together


Can you please help?
The answer is $3m^2+3m+1$ in the book that I saw
Thanks

First I will dramatically overcount. Then I will overcompensate for my overcounting. Then I will compensate for my overcompensation to reach the final answer. 

Imagine the garland as a fixed circle, with a total of $3+6m$ positions for flowers. Then the number of possible garlands is simply the number of ways to choose the $3$ positions of the white flowers. This is the binomial coeffient $$\binom{3+6m}{3} = \frac{(6m+1)(6m+2)(6m+3)}{6}$$
But we've agreed that garlands that can be rotated to be the same fixed garland are considered the same garland. Each garland can be rotated in $3+6m$ ways, so we have to divide our answer above by $3+6m$, to get
$$\frac{(6m+1)(6m+2)}{6}$$
This is N.F.Taussig's answer, but it's missing a little something. This is the one garland that remains the same when rotated by an angle less than $2\pi$. This special garland has the white flowers arranged in an equilateral triangle, and any rotations by $2\pi/3$ do not change the garland. There are $1+2m$ fixed versions of this garland, and we've counted them each $1/(3+6m)$ times. That means we've counted $\frac{1+2m}{3+6m} = 1/3$ of these garlands, when there is actually a full $1$ garland of this type. We need to add $2/3$ to our previous answer, to get our final answer of
$$\frac{(6m+1)(6m+2)}{6}+\frac{2}{3} \,= \,\boxed{6m^2 + 3m+1} \,\,\text{ garlands}$$
