Required the number of ways in which a number can be compounded of other numbers, different orders counting as different ways. Thus, 1 + 3 + 1 and 1 + 1 + 3 are to be considered as distinct ways of making 5. It will be obvious, on a little examination, that each number can be composed in exactly twice as many ways as the preceding number. Take 8 for instance. If every possible way of making 7 be written down, 8 may be made either by increasing the last component by a unit, or by annexing a unit at the end. Thus, 1 + 3 + 2 + 1 may yield 1 + 3 + 2 + 2, or 1 + 3 + 2 + 1 + 1: and all the ways of making 8 will thus be obtained; for any way of making 8, say _a_ + _b_ + _c_ + _d_, must proceed from the following mode of making 7, _a_ + _b_ + _c_ + (_d_-1). Now, (_d_-1) is either 0--that is, _d_ is unity and is struck out--or (_d_-1) remains, a number 1 less than _d_. Hence it follows that the number of ways of making _n_ is 2ⁿ⁻¹. For there is obviously 1 way of making 1, 2 of making 2; then there must be, by our rule, 2² ways of making 3, 2³ ways of making 4; and so on. { 1 + 1 + 1 { 1 + 1 + 1 + 1 { 1 + 1 { { 1 + 1 + 2 { { 1 + 2 { 1 + 2 + 1 1 { { 1 + 3 { { 2 + 1 { 2 + 1 + 1 { 2 { { 2 + 2 { 3 { 3 + 1 { 4 This table exhibits the ways of making 1, 2, 3, and 4. Hence it follows (which I leave the reader to investigate) that there are twice as many ways of forming _a_ + _b_ as there are of forming _a_ and then annexing to it a formation of _b_; four times as many ways of forming _a_ + _b_ + _c_ as there are of annexing to a formation of _a_ formations of _b_ and of _c_; and so on. Also, in summing numbers which make up _a_ + _b_, there are ways in which _a_ is a rest, and ways in which it is not, and as many of one as of the other.