Clearly, in one formula unit of Co(ClO_4)_2, there are 8 oxygen atoms. Agreed? In one formula unit of Co(ClO_4)_2*6H_2O, there are 14 oxygen atoms. Equivalently there are 14 oxygen atoms per formula unit of the solvate.
And in a 1*mol quantity of Co(ClO_4)_2*6H_2O there are thus clearly 14xx1*mol oxygen atoms. I use the mole as a number, just as I would use a "dozen" or a "score" or a "100", and of course the "mole" IS a number, i.e. N_A=6.022xx10^23*mol^-1.
And thus number of oxygen atoms in a "mole" of Co(ClO_4)_2*6H_2O,
=14*molxxN_A=14*cancel(mol)xx6.022xx10^23*cancel(mol^-1)
~=9xx10^24" oxygen atoms, a number as required. "
And in 3.42*mol, there are ~=3.1xx10^25" oxygen atoms."