Combining the Spins of 3 spin 1 particles

[rmiller70015]

Homework Statement

Find the normalized spin states for three identical non-interacting bosons where two have m_s = 1, and one has m_s = 0

Relevant Equations

None

I am having trouble with the normalization part.
To get a spin $|32>$ state I could have the following possibilities
$C_1|111110> + C_2|111011> + C_3|101111>$

This should be equivalent to
$C_1|11>|21> + C_2|11>|21> + C_3|10>|22>$
That is a spin 1 particle and a spin 2 particle that need to be combined.
So, to get the normalization coefficients would I just look these up in the Clepsch Gordon Table and that's all?

 

[kuruman]

I would be careful here. If you add three spins $S=1$ with $M_S=2$, is $| 32\rangle$ the only possibility?

 

[rmiller70015]

Thanks, I see what the problem is, also I was using bad representation of the spins, this is from chapter 7 problem 7 in Sakurai 2nd ed. and the way my professor told me to do the problem made it a lot easier.