In some of these questions, all you need to know is the number of atoms or molecules in a mole of a given molecule or compound. There is an Avogadro's number of atoms in a mole of atoms. Avogadro's number is equal to 6.022 x 10^23. Thus, moles of Ni can be determined as the ratio of the given number of atoms to Avogadro's number.
For example, question 1, part (i) can be solved as:
Moles of Ni = 2.15 x 10^17 / 6.022 x 10^23 = 3.57 x 10^-7 moles Ni
Similarly, atoms in 0.85 mole Ba = 0.85 x 6.022 x 10^23 = 5.12 x 10^23 atoms Ba.
In some of the cases (questions 3, 4, 5 & 8) a knowledge of molar mass is needed.
For example, in question 4 (part i), the atomic mass of Li is 7 g/mole and thus, 12.5 g Li is equal to 12.5/7 mole = 1.79 moles Li.
Provided with the mass of a compound, we can use its molar mass to figure out how many moles it is. Using Avogadro's number, we can convert the number of moles of the substance to the number of atoms/molecules.
Hope this helps.