Spin correlation problem

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Hi,
I'm having problems with this quantum mechanics problem. This is from a practice final I found online somewhere.
A particle of spin S has either spin S=0 or spin S=1. It decays into two electrons, and the spin of the two electrons is that of the original particle. The electrons come out along the x-axis. 50% of the time, electron 1 has s_z = +1/2 and electron 2 has s_z= -1/2, while the other 50% of the time electron 2 has s_z = +1/2 and electron 1 has s_z= -1/2



The question is what are the possible values of S and Sz consistent with this result? The answer is:
S = 1, S_z = 0
or
S = 0, S_z = 0



If, instead, the particle is equally likely to be in any of the 2S+1 eigenstates of S_z, how does the answer change?
S = 0, S_z = 0


Can someone explain this, at least the first part?

Thanks

Thanks.
 

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