# CHANGES

, the permutations or variations of any number of things, with regard to their position, order, &c; as how many changes may be rung on a number of bells, or how many different ways any number of persons may be placed, or how many several variations may be made of any number of letters, or any other things proposed to be varied.

*To sind out such number of changes,* multiply continually
together all the terms in a series of arithmetical
progression, whose first term and common difference
are each unity or 1, and the last term the number of
things proposed to be varied, thus 1 X 2 X 3 X 4 X 5
&c. till the last number be the proposed number of
things. For,

If there be only two things, as *a* and *b,* they admit
of a double order or position only; for they may be
placed either thus *ab* or thus *ba,* viz, .
If there be three things, *a, b,* and *c,* they

a | b | c |

a | c | b |

b | a | c |

b | c | a |

c | a | b |

c | b | a |

And if there be 4 things, each of them may be combined 4 ways with each order of the other three, that is 4 times 6 ways, or ways.

In like manner, the combinations of 5 things are of 6 things are &c.

So that if it be proposed to assign how many different ways a company of 6 persons may be placed, at table for instance, the answer will be 720 ways. Also the number of changes that can be rung on 7 bells, are changes.