34 ESSAY ON PROBABILITIES. 



Thus the number of permutations of 6 out of 9 is 

 opposite to 6 under 9> or 60480. 



2. By a combination is meant the same thing as a 

 permutation, except only that arrangement is no part 

 of the idea. Thus, of all the permutations of three out 

 of four in A, B, C, D, the following, A B C, B A C, 

 ACB, CAB, BCA, and CB A, are all the same 

 combination. Thus we have 



a permutation : a selection in a certain order. 



a combination : a selection without reference to order. 



To find the number of combinations, divide the 

 number of permutations by the product of ail the 

 numbers up to the number in each combination. Thus, 

 how many combinations can be made of 4 out of 

 12 ? The answer is found by 



dividing 12x11x10x9"! ,. , . 



by 1* 2x 3 x 4 j whlch S' Tes 495 ' 



The process may always be shortened by striking 

 common factors from the dividend and divisor. Thus, 

 if we wish to know the number of different hands which 

 can be held at whist (combinations of 13 out of 52) we 

 must 



divide 52. 51. 50. 49. 48. 47. 46. 45. 44. 43. 42. 41. 40. 

 by 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 



The shortest way is to decompose every number in 

 both into its factors, when it will appear that all the 

 factors of the divisor are found" among those of the di- 

 vidend; as follows: 



IS. 2.2. 17.3.5.5.2.7.7. 2,2.3.2.2. 47. 2.23. 5.3.3. 11.2.2. 



43. 2.7.3. 41. 2.2.2.5. 



1. 2. 3. 2.2. 5. 3.2. 7. 2.2.2. 3.3. 5.2. 11. 2.2.3. 13. ; 



and the factors which must be multiplied to give the 

 result are 



17. 7. 47. 23. 5. 43. 2.7. 41. 2.2.2.5. 

 or 17. 47. 23. 43. 41. 35. 560 giving 635013559600. 



When the number in each combination is more than 

 half of the whole number, the rule may be shortened. 



