A N A L Y S I S - A L C H E M Y|
A Five-Minute Course in Advanced Algebra
Potion recipes follow the mathematical laws of combinations or permutations.
Combinations deal with a number of objects taken so many at a time, regardless of order. For example, in the card game of Cribbage, pairs (cards of the same denomination) count two points. If you were dealt a pair of aces, you would score two points. Suppose you were dealt three aces: the ace of hearts, the ace of spades, and the ace of clubs. Letís abbreviate them as H (hearts), S (spades), C (clubs). You could make three different pairs -- HS, HC, and SC -- thus scoring six points instead of the three that you might casually think. This example shows three objects (the aces) taken two at a time (the pairs) regardless of order. In potion-making, combinations are the rule, with permutations being the rare exception. As you saw in the Art section, we had three reagents, which we combined two at a time, regardless of order, thus making Complex Potions.
Permutations are ordered sets of objects taken so many at a time. Suppose we wanted to arrange our Simple red (R), blue (B), and yellow (Y) Potions attractively in the first two slots of our Alchemistís pack. We could arrange them RB, RY, BR, BY, YR, and YB. These are permutations, which add up a lot faster than mere combinations. In permutations, RB is not the same as BR. In combinations it is.
For those who would like to delve further (or refresh their memories), the formula for combinations is
R! (N - R)!
where you have N objects taken R at a time. The exclamation point is a mathematical symbol for a factorial. Factorial N means all the numbers from one to N, multiplied together. With our alchemy of only three colors of reagents, itís easier just to count. Use a large number of possibilities, and the numbers mount up rapidly. Since the laws of probability are based in part on combinations, you can see why the odds are strongly against your winning the State Lottery!
The permutation formula is very similar to the combination formula, which is
(N Ė R)!