By definition, there are about `6*10^(23)` molecules in each mole of any substance. This number is called Avogadro number and is usually denoted as `N_A.` If we'll find the number of moles in `10` grams of nitric acid, we'll know the number of molecules by multiplying it by `N_A.`
Atoms of each type have its own mass. This mass, expressed in grams per mole, may be found in many sources including the periodic table. The values for `H,` `N` and `O` should be memorized and are about `1,` `14` and `16` grams per mole, respectively. Therefore the mass of one molecule of `HNO_3` is `1 + 14 + 3*16 = 63 (g/(mol)).`
Hence we have `(10 g)/(63(g/(mol))) approx0.16 mol` of nitric acid, and the number of molecules is about `6*10^(23)*0.16 approx 0.95*10^(23) = 9.5*10^(22).` This is the answer.