ALTERNATION

, or Permutation, of quantities or things, is the varying or changing the order or position of them.

As suppose two things a and b; these may be placed either thus ab or ba that is two ways, or 1 X 2. If there be three things, a, b, c, then the 3d thing c, may be placed three different ways with respect to each of the two positions ab and ba of the other two things, it may stand either before them, or between them, or after them both, that is, it may stand either 1st, 2d, or 3d; and therefore with three things there will be three times as many changes as with two, that is 1X2X3 or six changes with three things. Again, if there be four things a, b, c, d; then the fourth thing d may be placed in four different ways with respect to each of the six positions of the other three; for it may be set either 1st or 2d or 3d or 4th in the order of each position; consequently from four things there will be four times as many alternations as there are from three things; and therefore 1 X 2 X 3 X 4 = 24 is the number of changes with four things. And so on, always multiplying the last found number of alternations by the next number of things; or to find the number of changes for any number of things, as n, multiply the series of natural numbers 1, 2, 3, 4, 5, &c, to n, continually together, and the last product will be the number of alternations sought; so 1X2X3X4X5 - - - - n is the number of changes in n things.

So if, for example, it were required to find how many changes may be rung on 12 bells; it would be 1 X 2 X 3 X 4 X 5 X 6X7X8X9X10X11X12= 479001600, the number of changes. Now supposing there might be rung 10 changes in one minute, that is 10X12 or 120 strokes in a minute, or 2 strokes in each second of time; then, according to this rate, it would take upwards of 91 years to ring over all these changes on the 12 bells only. Also, if but two more bells were added, making 14 bells; then, at the same rate of ringing, it would require about 16575 years to ring all the changes on 14 bells but once over. And if the number of bells were 24, it would require more than 117000000000000000, years to ring all the different changes upon them!