COMBINATIONS AND PERMUTATIONS. 213 



planet appears ; so that the total variety of conjunctions is 

 2 s i 8 or 247. 



If an organ has twelve stops, we find in the thirteenth 

 line the numbers of combinations which we can draw, 



0, i, 2, 3, &c., at a time; the total number of modes of 

 varying the sound is no less than 2 12 i or 4095 m . If a 

 number be the product of n prime factors, we find in the 

 n+ I th line the numbers of divisors, being the product of 



1, 2, 3, or more of the prime factors; and the whole 

 number of divisors of th3 number is the sum of the 

 numbers in the line, subtracting unity, or 2" i. 



One of the most important scientific uses of the arith- 

 metical triangle, consists in the information which it gives 

 concerning the comparative frequency of divergencies from 

 an average. Suppose, for the mere sake of argument, 

 that all persons were naturally of equal stature of five 

 feet, but enjoyed during youth seven independent chances 

 of growing one inch in addition. Of these seven chances, 

 one, two, three, or more, may happen favourably to any 

 individual, but as it does not matter what the chances 

 are, so that the inch is gained, the question really turns 

 upon the number of combinations of o, i, 2, 3, &c., 

 things out of seven. Hence the eighth line of the triangle 

 give us a complete answer to the question, as follows : 

 Out of every 128 people- 

 Feet. Inches. 



One person would have the stature of 5 o 



7 persons ., 51 



21 persons 52 



35 persons 53 



35 persons 54 



21 persons 55 



7 persons 56 



i person 57 



m Bernoulli!, ' De Arte Conjectandi,' trans, by Maseres, p. 64. 



