Bernoulli III

Part Two.
The Doctrine of Permutations and Combinations.
“…there is no error into which even the most prudent and circumspect more frequently fall than the error that the logicians call the insufficient enumeration of the parts. As a result, the Art called Combinatorics should be judged, as it merits, most useful, because it remedies this defect of our minds and teaches us how to enumerate … so that we may be sure that we have omitted nothing that can contribute to our purpose.”
“On Permutations”
Definition: “The permutations of things I call the variations according to which, preserving the same multitude of things, their order and position are changed in different ways. Thus, if one asks in how many ways several things can be transposed or mixed with each other, so that one always takes them all and changes only their order or position, one is said to ask for all the permutations of these things.” In this section, Bernoulli uses letters to represent different numbers. Thus, of course, if the same letter is used twice, it is taken to be the same number.
First we look at the case where all objects to be “permuted” are different from each other.
The synthetic way: Starting from the very simple, we see that one thing, a, can only be arranged one way, whereas two things, a,and b, can be arranged two ways. Namely, ab and ba. With three letters, a,b,c, we find the possible arrangements by see that when we choose one letter to go in the first place there are only two possible arrangements for the other two (as we just saw). Thus, if a is first, b and c can be written in only two ways: ab, ba. The same if we choose b to go first with a and c. So, for a total of three letters there are 3x2=6 permutations: abc, acb, bac, bca, cab, cba. Similarly, with four letters, each of them can be ordered to take the first place with the other three varying their order (in 3x2=6 ways) and since the first place can contain four different letters, we see that the first position can be written in four different ways with the remaining positions being written in six different ways. Therefore, we see that four letters can be written in 4x3x2=24 ways. This rule can be repeated to find the number of permutations with 5 letters and so on. Here Bernoulli gives a rule for finding all the permutations of any number of letters. “If all the numbers from one following in natural order up to the given number of things inclusively are multiplied together, the product will reveal what was sought. For instance, if the given number of things is n, the number of permutations will be 1∙2∙3∙4∙5 etc up to n.” This is exactly our definition of n factorial! (or written: n!)


Next he examines permutations when some of the things to be permuted are the same.
Bernoulli demonstrates that if you have a group of letters, say, aaabcd, with one of the letters occurring multiple times, if you were to just take those letters (aaa) and try to rearrange them you would have the same outcome no matter what. Therefore, it is as if these like letters are only one letter. It must be noted that, were they different they could be permutated in 3x2=6 different ways. To take this into account when finding the totally permutations of all the letter in aabcd, we must make the number of permutations six times less from the total number of permutations that would occur if all number had been diverse. If the six letters were different we would have 1∙2∙3∙4∙5∙6 = 720 permutations. But 3 of the letters are alike! Therefore we decrease 720 by a multiple of 6 to get 120. In the same way, if b is repeated twice, as in: aaabbc, then then number of permutation is now twice smaller, or there are 60 permutations.
Bernoulli’s Rules: Rule 1. “If one of the things recurs more than once, the number of permutations that the given things would admit [produce] if all were different should be divided by the number of permutations that the similar things can undergo given their number.”
Rule 2. “Or, if there are several that repeat more often than once, the number of permutations that the given things would admit [produce] if all were different should be divided by the product of the numbers of permutations that each group of similar things could undergo individually according to it’s multitude [according to the number of things in each group].”
Here he gives an example using transpositions of Latin words. The first word he uses is Rome. By the first rule, the permutations of the letters in “Rome” are 1∙2∙3∙4=24 because all the letters are different. However, “Leopoldus” has 2 letters repeating or 2x2=4 repetitions. Therefore, according to the second rule, we find the total number of permutations that would occur were all the letters different. Since there nine letters in “Leopoldus” the number of permutations would be 1∙2∙3∙4∙5∙6∙7∙8∙9 = 362,880. Now dividing this number by the number of repetitions multiplied together gives us = 90,720. For “Studiosus,” we again follow the second rule because there are two u’s and three s’s. With 9 total letters in the word, this gives us = 30,240. He goes even further to construct the different permutations of Latin poems, or Bauhusian Verse (which will not be included here).
Combinations
Chapter IV. “To find the number of combinations of single exponents separately; and to show at the same time in how many combinations one or more designated things are found together or separately.”
Rule for finding the combinations of a given exponent.
“Let there be two arithmetic progressions, one descending from the number of things to be combined, the other ascending from1 and let the common difference in each progression be 1 and let there be as many terms in either progression as the exponent of the desired combination. Let the product of the terms of the first progression be divided by the product of the terms in the second progression. The quotient will be the desired number of combinations sought.”
For example: If you have ten things, and you want to see how many ways you can take 4 things from 10, the formula is: = 5040/24 = 210.
This rule is used in Part Three as we shall now see.