and so on.

205. Each of these combinations may be written in several different orders; thus, abcd may be disposed in any of the following ways:

abcdacbdacdbabdcadbcadcb
bacdcabdcadbbadcdabcdacb
bcadcbadcdabbdacdbacdcab
bcdacbdacdbabdcadbcadcba

of which no two are entirely in the same order. Each of these is said to be a distinct permutation of abcd. Considered as a combination, they are all the same, as each contains a, b, c, and d.

206. We now proceed to find how many permutations, each containing one given number, can be made from the counters in another given number, six, for example. If we knew how to find all the permutations containing four counters, we might make those which contain five thus: Take any one which contains four, for example, abcf in which d and e are omitted; write d and e successively at the end, which gives abcfd, abcfe, and repeat the same process with every other permutation of four; thus, dabc gives dabce and dabcf. No permutation of five can escape us if we proceed in this manner, provided only we know those of four; for any given permutation of five, as dbfea, will arise in the course of the process from dbfe, which, according to our rule, furnishes dbfea. Neither will any permutation be repeated twice, for dbfea, if the rule be followed, can only arise from the permutation dbfe. If we begin in this way to find the permutations of two out of the six,

a b c d e f

each of these gives five; thus,

a gives ab ac ad ae af

b ... ba bc bd be bf

and the whole number is 6 × 5, or 30.