Elements of Algebra

CHAP. XI. OF ALGEBRA. 117 first, and the four others will admit of twenty-four transpositions; so that the whole number of transpositions will be $5 \times 24 = 120 = 5 \times 4 \times 3 \times 2 \times 1$. 356. Consequently, however great the number of letters may be, it is evident, provided they are all different, that we may easily determine the number of transpositions, and, for this purpose, may make use of the following Table: Number of Letters. Number of Transpositions. 1 $1=1$. 2 $2.1=2$. 3 $3.2.1=6$. 4 $4.3.2.1=24$. 5 $5.4.3.2.1=120$. 6 $6.5.4.3.2.1=720$. 7 $7.6.5.4.3.2.1=5040$. 8 $8.7.6.5.4.3.2.1=40320$. 9 $9.8.7.6.5.4.3.2.1=362880$. 10 $10.9.8.7.6.5.4.3.2.1=3628800$. 357. But, as we have intimated, the numbers in this Table can be made use of only when all the letters are different; for if two or more of them are alike, the number of transpositions becomes much less; and if all the letters are the same, we have only one arrangement: we shall therefore now shew how the numbers in the Table are to be diminished, according to the number of letters that are alike. 358. When two letters are given, and those letters are the same, the two arrangements are reduced to one, and consequently the number, which we have found above, is reduced to the half; that is to say, it must be divided by $2$. If we have three letters alike, the six transpositions are reduced to one; whence it follows that the numbers in the Table must be divided by $6=3.2.1$; and, for the same reason, if four letters are alike, we must divide the numbers found by $24$, or $4.3.2.1$, &c. It is easy therefore to find how many transpositions the letters $aaabbc$, for example, may undergo. They are in number $6$, and consequently, if they were all different, they would admit of $6.5.4.3.2.1$ transpositions; but since $a$ is found thrice in those letters, we must divide that number of transpositions by $3.2.1$; and since $b$ occurs twice, we must again divide it by $2.1$: the number of trans-