- If a number of counters, distinguished by different letters, be placed on the table, and any number of them, say four, be taken away, the question is, to determine in how many different ways this can be done. Each way of doing it gives what is called a combination of four, but which might with more propriety be called a selection of four. Two combinations or selections are called different, which differ in any way whatever; thus, abcd and abce are different, d being in one and e in the other, the remaining parts being the same. Let there be six counters, a, b, c, d, e, and f; the combinations of three which can be made out of them are twenty in number, as follow:
| abc | ace | bcd | bef |
|---|---|---|---|
| abd | acf | bce | cde |
| abe | ade | bcf | cdf |
| abf | adf | bde | cef |
| acd | aef | bdf | def |
The combinations of four are fifteen in number, namely,
| abcd | abde | acde | adef | bcef |
|---|---|---|---|---|
| abce | abdf | acdf | bcde | bdcf |
| abcf | abef | acef | bcdf | cdef |
and so on.
- Each of these combinations may be written in several different orders; thus, abcd may be disposed in any of the following ways:
| abcd | acbd | acdb | abdc | adbc | adcb |
|---|---|---|---|---|---|
| bacd | cabd | cadb | badc | dabc | dacb |
| bcad | cbad | cdab | bdac | dbac | dcab |
| bcda | cbda | cdba | bdca | dbca | dcba |
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.
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