How many necklaces can be made from 18 beads of 3 different colours. How many different necklaces can be made from beads. If we have


*

*10 green beads

*5 red beads

*3 yellow beads


And we will use all the beads for creation of necklace.
Disclaimer: Not a homework question, just preparing for test
 A: There are $\frac{18!}{10!\times 5!\times 3!}$ different ways to arrange the beads in a row.
To see this, start by labelling the beads to make them all different, and now there are $18!$ ways to order the different beads. However, every arrangement of unlabelled beads corresponds to $10!\times 5!\times 3!$ arrangements of the labelled beads (you can rearrange the green labels in $10!$ ways, etc.). 
But we want the number of necklaces, not the number of ways to arrange the beads in a row. Now each necklace corresponds to $18$ ways of arranging the beads in a row - break the necklace at any point and start the row there. These $18$ arrangements are always all different (this is not necessarily true for a general necklace problem, but it is here). This is because if two different places, $k$ apart, to break the necklace gave the same arrangement, there would always be another red bead $k$ places after every red bead, and since $5$ and $18$ are coprime this is not possible.
Thus we have to divide the number of arrangements by $18$ to get the number of necklaces, which is therefore $\frac{17!}{10!\times 5!\times 3!}$
A: The answer is $$\frac{18!}{10!.5!.3!} =2450448$$ Here is how we count the necklaces.
For the first bead you have $18$ choices, for the second one you have $17$ choices, and so forth ...until the last bead for which you have only $1$ choice.
That makes it $18!$ if all the beads were different. Since we can rearrange the  $ 10$ green beads in $10!$ ways and rearrange the $5$ red beads in $5!$ ways and the $3$ yellow beads in $3!$ ways, we divide the $18!$ by $(10!5!3!)$ and get the result.  
