212 THE PRINCIPLES OF SCIENCE. 



the number of ways in which I can select i, 2, 3, 4, 5, 6, 

 7, and 8 things, out of 8 things. In general language, if 

 I wish to know in how many ways m things can be 

 selected in combinations out of n things, I must look 

 in the n + I th line, and take the m + i th number, counting 

 from the left, as the answer. In how many ways, for 

 instance, can a sub-committee of five be chosen out of a 

 committee of nine. The answer is 126, and is the sixth 

 number in the tenth line ; it will be found equal to 



, which our previous formula (p. 206) would 

 1.2.3.4.5 



give. 



The full utility of the figurate numbers will be more 

 apparent when we reach the subject of probabilities, but I 

 may give an illustration or two in this place. In how 

 many ways can we arrange four pennies as regards head 

 and tail \ The question amounts to asking in how 

 many ways we can select o, i, 2, 3, or 4 heads out of 4 

 heads, and the fifth line of the triangle gives us the 

 complete answer, thus 



We can select No head and 4 tails in i way. 

 i head and 3 tails in 4 ways. 



2 heads and 2 tails in 6 ways. 



3 heads and i tail in 4 ways. 



4 heads and o tail in i way. 



The total number of different cases is 16, or 2 4 , and 

 when we come to the next chapter, it will be found that 

 these numbers give us the respective probabilities of all 

 throws with four pennies. 



I gave in p. 205 a calculation of the number of ways in 

 which eight planets can meet in conjunction ; the reader 

 will find all the numbers detailed in the ninth line of the 

 arithmetical triangle. The sum of the whole line is 2 8 or 

 256 ; but we must subtract a unit for the case where no 

 planet appears, and 8 for the 8 cases in which only one 



