ix. J COMBINATIONS AND PERMUTATIONS. 181 



only through the inherent imperfections of our symbols 

 and modes of calculation. Signs must be used in some 

 order, and we must withdraw our attention from this order 

 before the signs correctly represent the relations of things 

 which exist neither before nor after each other. Now, it 

 often happens that we cannot choose all the combinations 

 of things, without first choosing them subject to the 

 accidental variety of order, and we must then divide by 

 the number of possible variations of order, that we may 

 get to the true number of pure combinations. 



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 x 25 x 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^?, 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, 



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It is apparent that we need the doctrine of combina- 

 tions 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 x 29 x 28 x 27 x 26 selections; 

 but as the actual character of the members of the committee 

 will not be affected by the accidental order of their selec- 

 tion, we divide by 1x2x3x4x5, and the possible 

 number of different committees will be 142,506. Similarly 

 if we want to calculate the number of ways in which uhe 

 eight major planets may come into conjunction, it is evi- 

 dent 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 



