i.  One way to approach this problem is to do Part Two first by trial and error and then count the Fischer projections you have drawn in order to find the answers to Part One.  The disadvantage of this plan is there is no way of knowing with certainty if you have drawn all Fischer projections.  The mathematical concept factorials provides a better method.  Let us first consider a simple non-chemistry example.  Four kids, A, B, C, and D, go to the park to play.  There is a teeter-totter (seesaw) in the park and the kids decide to play on it.  In how many different ways could the four kids occupy the four seats of the teeter-totter?

Using the factorials, the answer is 4! = 1 x 2 x 3 x 4 = 24.

Let us now consider lactic acid.  The lactic acid molecule has one chiral center.  Therefore, a Fischer projection representing either enantiomer of lactic acid has the following bond skeleton around the chiral center.

There are four ligands to be placed on the four sites: CH3, CO2H, OH, and H.  The number of permutations in which the four ligands can be distributed among the four sites is 4! = 1 x 2 x 3 x 4 = 24.  Some of these 24 Fischer projections represent (R)-lactic acid and others (S)-lactic acid.  The mirror image of a Fischer projection representing (R)-lactic acid represents (S)-lactic acid and vice versa, implying that twelve of the 24 Fischer projections represent (R)-lactic acid and the other twelve (S)-lactic acid.  Thus, the maximum number of Fischer projections that can be drawn to represent (R)-lactic acid is twelve.

2.  Use the following stepwise procedure to generate the twelve Fischer projections that represent (R)-lactic acid.

Step 1:  Draw four Fischer projection skeletons.

Step 2: Place one of the four ligands on site 1 of each skeletons without repeating any ligands.

Step 3: Complete each Fischer projection so that the chiral center in it has R configuration.  (To make the process easy, start by placing H on site 3 of 1, 2, and 3.

Step 4: Generate two more Fischer projections from each of the above Fischer projections, using the second operation that does not change the absolute configuration at the chiral center, keeping the ligand on site 1 fixed.

ii.  The following flow chart provides a stepwise approach to solving this problem.

*If two or more chiral centers in each formula bear the same set of ligands, one must be carful in identifying corresponding chiral centers.  For example, consider Fischer projections 1 and 2.

In each formula, the chiral centers are marked with asterisks.  Notice that the two chiral centers in each formula bear the same ligands; each chiral center bears a i) H atom, ii) OH group, iii) CO2H group, iv) CH(OH)CO2H group.  Determine the absolute configuration at each chiral center in 1 and 2.

It looks as if the corresponding chiral centers in 1 and 2 do not have the same absolute configuration.

However, rotate 1 by 180° in either direction without lifting it off the plane of the screen, an operation that does not change the absolute configurations of chiral centers in a Fischer projection, and compare the resultant Fischer projection (1a) with 2.

Notice that, in 1a and 2, the corresponding chiral centers have the same absolute configuration.  Alternatively, 1a, which represents the same molecule as does 1, is superimposable on 2.  Therefore, 1 and 2 must represent the same molecule.

The two chiral centers in each of these two Fischer projections bear the same ligands: H atom, OH group, CO2H group, and CH(OH)CO2H group.  However, in this case, it is obvious that the corresponding chiral centers do not have the same absolute configurations.  Therefore, the formulas are not identical.  Since not all corresponding chiral centers have opposite absolute configurations, the formulas represent diastereomers.

The two chiral centers in each of these two Fischer projections bear the same ligands: H atom, OH group, CO2H group, and CH(OH)CO2H group.  However, in this case, it is obvious that the corresponding chiral centers do not have the same absolute configurations.  Therefore, the formulas are not identical.  Since not all corresponding chiral centers have opposite absolute configurations, the formulas represent diastereomers.

The two chiral centers in each of these two Fisher projections bear the same ligands: H atom, Br atom, CH3 group, and CHBrCH3 group.  The two formulas appear to have opposite absolute configurations at the corresponding chiral centers.

However, rotate the first one by 180º in either direction without lifting it off the plane of the screen and compare the resultant formula with the second one.

Notice that in the two formulas the corresponding chiral centers have the same absolute configuration.  Therefore, the two formulas are identical.

The two chiral centers in each of these two Fischer projections bear the same two ligands: H atom, Br atom, CH3 group, and CHBrCH3 group.  However, in this case, it is obvious that the corresponding chiral centers in the two formulas have the same absolute configuration.  Therefore, the two formulas are identical.