1 8 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



1.402 Given n different elements. The number of ways they can be 

 divided into m specified groups, with xi, x 2 , , x m in each group respec- 

 tively, (xi + xa + + x m ) = n is 



nl 



e.g., n = 6, m = 3, xi = 2, x 2 = 3, x* = i : 



(12) (345) (6) (13) (245) (6) X 6 = 60 



(23) (145) (6) (24) (135) (6) 



(34) (125) (6) (35) (124) (6) 



(45) (123) (6) (25) (234) (6) 



(14) (235) (6) (15) (234) (6) 



1.403 Given n elements of which x\ are of one kind, x? of a second kind, 

 ....... , x m of an mth kind. The number of permutations is 



X m 



%i + x 2 + ...... + x m = n. 



1.404 Given n different elements. The number of ways they can be permuted 

 among m specified groups, when blank groups are allowed, is 



(m + n - i) ! 



(m-i)I 

 e.g., n = 3, m = 2 : 



(123,0) (132,0) (213,0) (231,0) (312,0) (321,0) (12,3) (21,3) (13,2) (31,2) (23,1) 

 (32,1) (1,23) (1,32) (2,31) (2,13) (3,12) (3,21) (0,123) (0,213) (0,132) (0,231) 

 (0,3 1 2) (0,321) = 24 



1.405 Given n different elements. The number of ways they can be permuted 

 among m specified groups, when blank groups are not allowed, so that each group 

 contains at least one element, is 



n\(n- i)l 



(n m)\(m i)l 

 e.g., n = 3, m = 2 : 

 (12,3) (21,3) (13,2) (31,2) (23,1) (32,1) (1,23) (1,32) (2,31) (2,13) (3,12) (3,21) = 12 



1.406 Given n different elements. The number of ways they can be combined 

 into m specified groups when blank groups are allowed is 



m n 

 e.g., n = 3, m = 2 : 



(123,0) (12,3) (13,2) (23,1) (1,23) (2,31) (3,12) (0,123) = 8 



1.407 Given n similar elements. The number of ways they can be combined 

 into m different groups when blank groups are allowed is 



