# 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
will admit of 6 variations = 1 X 2 since as in the margin, and no more; three each of the three may be combined there different ways with each of the other two. |

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.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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