The Universal Art of Music, Volume II (1650) - Page 17

TABLE I. Combinatorics.


1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800
11	39,916,800
12	479,001,600
13	6,227,020,800
14	87,178,291,200
15	1,307,674,368,000
16	20,922,789,888,000
17	355,687,428,096,000
18	64,023,737,057,280,000
19	121,645,100,408,832,000
20	2,432,902,008,176,640,000
21	51,090,942,171,709,440,000
22	1,124,000,727,777,607,680,000
23	25,852,016,738,884,976,640,000
24	620,448,401,733,239,439,360,000

Permutations, and so on into infinity.

You see, therefore, how for any number of things placed in their natural order, the number of permutations original: "mutationum" suitable to those things is produced from the multiplication of the numbers performed in sequence among them. Thus, 10 things can be combined among themselves 3,628,800 times, and 20 things 2,432,902,008,176,640,000 times. These figures can scarcely be believed, unless a mathematical demonstration convinced our intellect.

PROPOSITION II original: "Lemma II".

Permutations of a five-letter name original: "pentagrammati" in which two letters are the same.

Whenever similar things occur—or, what is the same thing, the same items are placed several times among the things to be combined—the combination of the whole divided by the number of combinations of the similar items (as proposed in the preceding table) will give the required number of combinations.

Suppose the name MARIA is proposed. Since it has two similar letters—or, what is the same thing, the letter A is placed twice—we ask how many times it can be combined. Since the name is a pentagrammaton A word consisting of five letters. or five-letter word, if all the letters were different, they would have 120 combinations as shown in the preceding table. Let this number be divided by the number of combinations for two items The factorial of 2, which is 2., as the table shows; it will give a quotient of 60. And that is how many times any 5-letter name can be combined if it has two letters that are similar or the same, as in the present name with the two As.

Of a three-letter name original: "trigrammati".

Again, let a trigrammaton A word consisting of three letters. or three-letter name be given in which two letters are placed twice, as in the name ALA, and someone wishes to know how many times it can be combined. He will achieve his wish if he divides the combination of the whole (namely the combination of three things, which is 6) by the number of combinations for two items (which is 2). The result will be 3, which is the required combination of the proposed name.