Rule 1: Always...

Rule 1: Always select the first white key ! The 'Sa'.
Rule 2: Always select the Pa key. This is a convenient midpoint of the octave, sort of.
Rule 3: Select one of the two Ma keys (Ma1 or Ma2 - note that one of them is black and the other one is white) Once selected, this key is your 'Ma'.
Rule 4: Select ANY two keys out of the four keys in the lower tetrachord. (From Keys 2, 3, 4 and 5) Once selected, the first of these two keys will be your 'Ri' and the second your 'ga'.
Rule 5: Select ANY two keys out of the four keys in the upper tetrachord. (From keys 9, 10, 11 and 12) Once selected, the first of the two keys will be your 'dha' and the second will be your 'ni'. This rule is exactly like Rule 4.
Once all the seven keys are chosen, you have your complete sa ri ga ma pa dha ni.
Let us see how many Melakartas or scales we can build this way. By Rule 4, you can choose two keys out of four in SIX different ways going by the elementary combination theory. Similary, going by Rule 5, we can choose two keys out of four in SIX different ways. By Rule 3, you can choose one key out of two in TWO different ways. So we get
SIX times SIX times TWO = Seventy Two Melakartas or Melakarta ragams.
And they are all unique.
By definition, the Melakarta Ragams are symmetric with respect to going up in octave or down. Saying the same thing more technically, in Melakarta Ragams, the Arohanam and the Avarohanams are simply reversed. The sequence Sa ri ga ma pa dha ni is Arohanam. The reversed sequence Sa ni dha pa ma ga ri is Avarohanam. The Melakarta Ragams are also called 'Sampoorna ragams' or Complete ragams.
Interestingly, even the Melakarta selection algorithm allows us to choose all seven white keys, the same as the Western C Major scale. In Karnatic music, we call the resulting Melakarta ragam as Shankarabharanam. (You may have even heard of this ragam) In Hindustani music, the set of all white keys is called the 'Bilaval thaat', one of the major building blocks of Hindustani musical system.
Let us now go back to Table II and see why notation 1 makes sense. For example, you can pick up any two keys from the keys 2, 3, 4 and 5 and still call the first one of those as Ri and the second one as Ga. If you chose keys 2 and 5 then, you will sing out 'ri' when you strike key 2 and 'ga' when you strike key 5. On the other hand, if you chose keys 3 and 4 you will say 'ri' for key 3 and 'ga' for key 4. Finally if you chose keys 2 and 3, then key 3 will be a 'ga' (and not 'ri') in this situation. The rule is, the first key used among these four keys is a 'ri' and the second one is 'ga' no matter which absolute position the keys are located at. Keys 3 and 4 have the dubious honor of being a ri or a ga depending on the situation. These arguments are also valid in the upper tetrachord and in the choice of 'dha' and 'ni'. Now perhaps we can understand why three keys were designated as 'ri' or 'ga' or 'dha' or ni.
A caveat. I am using the word 'Ragam' in a loose sense here. A Ragam is not just a scale or a bunch of keys - it is more than that. Remember, I told you over and over and over that microtones are everything in Indian classical music and keys in a keyboard are simply digitized approximations. The seven white keys alone are not enough to give the resulting music the flavor of ragam 'Shankarabharanam' - it is those seven keys PLUS all the associated microtones (I know, I am being vague, but there is no simple way to get around it !) which constitute the 'Shankarabharanam' ragam. In fact, you may hear shades of Shankarabharanam when someone plays the Western C Major or Hindustani Bilaval. But the 'shades' are different for C Major and Bilaval and Shankarabharanam. C Major does not have any gamakam, Bilaval has some and Shankarabharanam has another set of gamakams. It is important to listen to some music and figure out if you can identify an artiste go through gamakams. A simple rolling of the tongue, subtle jumps and modulation or vibrattos are all indicative of gamakams.
Also, if you are the type that questions authority, you may equally well question the Melakarta selection rules. Why should we include Pa always and why can't we include BOTH the Ma1 and Ma2 keys in the same scale ? In Hindustani music there are ragams which use both the Ma keys, although it is a no-no in Karnatic. (once you become more advanced you will see that even in Karnatic music some pieces use both the Mas)
Finally, we should notice a fundamental difference between the Western system of scale building compared to the Melakarta scheme. In the Western classical music, you started off on a specific key, used the algorithm to generate the next key, which in turn led you to the third key of the scale and so forth. You sequentially generated the keys one after another by just shifting a whole tone or half a tone. By a curious coincidence even the Ilikkramam algorithm in Silappadhikaram is a similar 'Mode shifting' or 'tone shifting' algorithm. By contrast, the Melakarta scheme is a brutally mathematical scheme where you selected 7 keys out of a possible 12 keys, subject to certain constraints - here you figured out the frequency relationship between the keys much later. One important consequence: In the Western scale system, the keys in a scale are not more than a 'whole tone' apart, i. e, in any Major or Minor scale, you 'skip' at the maximum just one key. Whereas in Melakarta scheme, you can choose Key 1, Key 2, Key 3, Key 7, key 8, key 11 and key 12 by the algorithm. (This corresponds to Ragam Raghupriya) Notice the big gap between key 3 and key 7 (between the 'ga' and 'ma') where we skipped over three keys (This amounts to skipping two whole tones or four semitones). Also, we skipped two keys between 'pa' and 'dha'. (keys 8 and 11) Such large 'Intervals' ('Interval' is yet another musical term !) can produce 'unpleasant' listening experience. And although Raghupriya is a legitimate Ragam, it is about as popular as rain during a picnic.