36 



ESSAY ON PROBABILITIES. 



D A A, that is nine where A is repeated twice, and 

 A A A, one, where A is repeated three times. Ten 

 more are those in which B is repeated twice, &c. 



4. The number of combinations in which there are 

 repetitions must he determined without rule, in every 

 particular case. Suppose I wish to know how many com- 

 binations, including repetitions, can be made of four 

 out of seven. Let the seven be A, B, C, D, E, F, G. 



I. The number, without repetition, is 35. 



II. Those in which A is repeated twice, and no other, 

 are evidently as many as the number of pairs in B, C, 

 D, E, F, and G, that is, J 5. Thus we have A A B C, 

 A A C D, &c. There are as many in which B is re- 

 peated twice, &c., so that 7 X 15 or 105, is the num- 

 ber in which one only is repeated twice. 



III. The number in which A is repeated three times 

 is evidently 6, and the number in which one or other is 

 repeated three times only, is 6 X 7 or 42. 



IV. The number in which one is repeated four 

 times, is 7. 



V. The number in which two are repeated twice, is 

 as many as the number of pairs in 7j or 21. 



Consequently, the whole number is made up of 35 

 105, 42, 7, and 21, or it is 210. 



We have not so much to do with combinations allow- 

 ing repetition, as with permutations of the same kind. 



To avoid the perpetual occurrence of long phrases 

 for simple ideas, I shall use the following abbrevia- 

 tions: By P [4,20 } is meant the number of per- 

 mutations of 4 out of 20, without repetition : by P P 

 { 4,20 ] the same with repetition. By C [ 4,20 ] 

 is meant the number of combinations of 4 out of 20, 

 without repetition : by C C [ 4,20 j the same with 

 repetition. Again, by [4,20], as in page 15., is meant 

 the product of all numbers from 4 to 20, both inclu- 

 sive ; and, by 7 10 , as in algebra, is meant ten sevens 

 multiplied together. Thus, a reference to the preceding 

 rules will show the following : 



