D is the correct answer and here's an explanation:
Lets use b as a starting point, since you are sure that this is true. Now, what makes a mole so special that it is the exact amount of substance where the formula weight equals the mass in grams? First lets consider what formula weight is.
Atoms consist of protons, neutrons and electrons. Since the mass of electrons is negligible compared to the mass of protons and neutrons, we can ignore these for our purposes. Protons and neutrons have a mass of one atomic mass unit (amu or dalton). For simplicity's sake, lets consider helium-4, a single isotope. One atom of 4He consists of two protons and two neutrons. Therefore each atom has a mass of 4 amu and thus the formula weight of 4He is 4. From definition b, we know that one mole of 4He would weigh 4 g. Now how do we find how many He atoms is in 4g?
Well, we need a conversion factor (lets call it N) so that:
4 g x (N amu / g) = 4 amu
In other words, we need to find the weight of 1 gram in amu. It turns out that:
6.02x1023amu = 1g
But, since protons and neutrons have a mass of 1 amu, you can say that:
6.02x1023 protons = 1g
and
6.02x1023 neutrons = 1g
This number, like many other constants, is so important that it has been given a name, Avogadro's number (note: the definition of Avogadro's number is slightly different, but it is basically the number of protons in a gram).
With this in mind, we can now see that a and c are equivalent since Avogadro's number = 6.02x1023. And we can also see that a and c are true. Why? Well, the formula weight is basically just the sum of the number of protons and neutrons in a molecule/formula unit of a substance.* So, if you have a molecule of water, it has ten protons and eight neutrons. Therefore Avogadro's number of water molecules has 10x(6.02x1023) protons and 18x(6.02x1023) neutrons. Therefore the weight of protons in 6.02x1023 molecules of water is 10g and the weight of neutrons in this amount of water is 8g, giving a total mass of 18g.
*Basically for covalent substances you have molecules, but for ionic substances, like NaCl, molecules of NaCl don't exist since Na and Cl ions are bound in a repeating lattice. So, chemists call the simplest ratio of Na to Cl ions in the lattice a "formula unit." Basically, it's the same concept as a molecule of NaCl, but chemists call it a "formula unit" because it isn't correct to say that there are molecules of NaCl.