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It's easy to see that for addition of 2 angular momenta l1 and l2 , the space l1 m1 , l2 m2 is equivalent to the space of l1 l2 l m (where l is the total angular momentum).

Counting the total number of states is usually a convenient way to make sure you got the addition right.

But what about the addition of 3 angular momenta? consider for example, l1,l2,l3 all equal to 1.

It's easy to count the total number of states: 3X3X3=27.

Adding the momenta we can get l=0,1,2,3 and so the total number of states is 1+3+5+7=16.

So what happened to 27-16=11 missing states? There must be some quantum number to distinguish between them, right?