Suppose that we wish to determine the number of ways in which we can select a group of three letters out of the alphabet, without allowing the same letter to be repeated. At the first choice we can take any one of 26 letters; at the next step there remain 25 letters, any one of which may be joined with that already taken; at the third step there will be 24 choices, so that apparently the whole number of ways of choosing is 26 × 25 × 24. But the fact that one choice succeeded another has caused us to obtain the same combinations of letters in different orders; we should get, for instance, *a*, *p*, *r* at one time, and *p*, *r*, *a* at another, and every three distinct letters will appear six times over, because three things can be arranged in six permutations. To get the number of combinations, then, we must divide the whole number of ways of choosing, by six, the number of permutations of three things, obtaining (26 × 25 × 24)/(1 × 2 × 3) or 2,600. It is apparent that we need the doctrine of combinations in order that we may in many questions counteract the exaggerating effect of successive selection. If out of a senate of 30 persons we have to choose a committee of 5, we may choose any of 30 first, any of 29 next, and so on, in fact there will be 30 × 29 × 28 × 27 × 26 selections; but as the actual character of the members of the committee will not be affected by the accidental order of their selection, we divide by 1 × 2 × 3 × 4 × 5, and the possible number of different committees will be 142,506. Similarly if we want to calculate the number of ways in which the eight major planets may come into conjunction, it is evident that they may meet either two at a time or three at a time, or four or more at a time, and as nothing is said as to the relative order or place in the conjunction, we require the number of combinations. Now a selection of 2 out of 8 is possible in (8 . 7)/(1 . 2) or 28 ways; of 3 out of 8 in (8 . 7 . 6)/(1 . 2 . 3) or 56 ways; of 4 out of 8 in (8 . 7 . 6 . 5)/(1 . 2 . 3 . 4) or 70 ways; and it may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers of ways are 56, 28, 8, and 1. Thus we have solved the whole question of the variety of conjunctions of eight planets; and adding all the numbers together, we find that 247 is the utmost possible number of modes of meeting. In general algebraic language, we may say that a group of *m* things may be chosen out of a total number of *n* things, in a number of combinations denoted by the formula (*n* . (*n*-1)(*n*-2)(*n*-3) .... (*n* - *m* + 1))/(1 . 2 . 3 . 4 .... *m*) The extreme importance and significance of this formula seems to have been first adequately recognised by Pascal, although its discovery is attributed by him to a friend, M. de Ganières.[99] We shall find it perpetually recurring in questions both of combinations and probability, and throughout the formulæ of mathematical analysis traces of its influence may be noticed.