CHANGES, in arithmetic, the varia tions or permutations of any number of things, with regard to their position, order, &c. The method of finding out the number of changes, is by a continual multiplication of all the terms in a series of arithmetical progressionals, whose first term, and common difference, is unity, or 1; and last term the num ber of things proposed to be varied, viz. 1 X2x3x4x 5 X6x7,&c.aswill appear from what follows : 1. If the things proposed to be va ried are only two, they admit of a dou ble position, as to order of place, and no more Thus, 1 X 2.
2. And if three things are proposed to be varied, they may be changed six seve ral ways, as to their order of places, and no more.
For, beginning with 1, there ? 1 . 2 . 3 will be ..... .... ........ ..... ...... 5 1 . 3 . 2 Next, beginning with 2, there? 2 .1 .3 will be ..... ........... ....... . ....... 5 2 .3 . 1 Again, beginning with 3, it ? .1 .2 will be 3 3.2.1 Which, in all, make 6, or 3 times 2 ;viz.
1 x 2 x 3 = 6.
3. Suppose 4 things were supposed to be varied, then they admit of 24 several changes, as to their order of different places.
) 1.2.3.4 t For, beginnning the order 1 .2 . 4 .3 with 1, it will be........... 1 . 3 . 2 . 4 1.3.4.2 Here are six different 1.4.2.3 changes............. ......... ... 1 . 4 .3 . 2 And for the same reason there will be 6 different changes when 2 begins the order, and as many when 3 and 4 begin the order ; which, in all, is 24 = I X 2 They may thus be continued on to any assigned number. Suppose to 24, the num ber of letters in the alphabet, which will admit of 620448401733239439360000 se veral variations.
Since on 12 bells there would be, by the table, 479001600 changes : suppose 10 changes to be rung in a minute, that is 10 x 12, or 120 strokes in a minute, it would even then require upwards of 90 years to ring over all the changes on the 12 bells.