I can't help with explaining this specifically but I did the following, I developed |B_{01}> into (1/2)^{1/2} (|00> - |11>), we get
1/8 || |000> - | 110> - | 001> + |111>||^2 (all states are orthonormal) = 1/8 * (1+1+1+1) = 4/8 = 1/2
1/8 || |000> - | 110> - | 001> + |111>||^2 (all states are orthonormal) = 1/8 * (1+1+1+1) = 4/8 = 1/2
