COMBINATIONS AND PERMUTATIONS. 205 



number of ways of choosing 1326x25x24. 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. Thus the true number of combinations 



.-,-, -, 24x23x22 

 will be - 1 - , or 2024. 

 1x2x3 



It is apparent that we need the doctrine of permuta- 

 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 30x29 x28 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 num- 

 ber 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 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 

 the relative order or place in the conjunction, we require 

 the number of combinations. Now a selection of 2 out of 8 



Of- R H f\ 



is possible in - - or 28 ways ; of 3 out of 8 in 



or 56 ways ; of 4 out of 8 in -|ILJ2 O r 70 ways ; and it 



1.2.3.4 



may be similarly shown that for 5, 6, 7, and 8 planets, 

 meeting at one time, the number of ways is 56, 28, 8 

 and i. 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 



