30 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
(h l)st group containing (m — Jc + h— 1) A combinations, consisting of 
those which have in common the first (k — h) only of the numbers 
of (n\m \ k). 
a jS 
(/c + l)th and last group containing (m— 1)* combinations, consisting of 
those which do not contain the first number in (n\m\ k). 
a J8 
Let 
(n\m\tc\Ic-h)(c m ), (* = 0 , 1 , 2 , . . . (m-k + h- 1 ) - 1 ) 
a 0 1 1 h 
represent the combinations of the (/& + l)st group. By giving h in this 
the values from 0 to k, it will represent the combinations of each group. 
Let the first combinations in the 1st, 2nd, . . . (& + l)th groups, ar- 
ranged in this order, be represented by 
(n | m | k ), (n \ m \ k ), . . . (n\m\k) i 
Po Pi a Pk 
respectively, and let 
p — 1 k 
a FT Tal 
0 0 
(n \m\ k ) 
a p h +i 
(n\m\k\k-h)(c W ) 
a P 1 1 
(p = (m-k + h — 1 ) ft ) 
represent the determinant whose elements along the principal diagonal 
arranged in the above order are 
(n | m | k) 
a 1 
(n\m\ k) 
a 2 
(?i | m | k) 
a \ 
(n | m k) 
a 1 
> 
(■ n | m | k) 
a 2 
(n\m\k) 
a \ 
Then the determinant 
(n | m | k ) 
p - 1 k 
d^a in III 
0 0 
(n | m) 
a 
(b< k - h) )(+ ) )(n\m\ k ) 
a P h +i 
A (m_1 k (C) 
where (b^ ll) ) represents any set of (k — h) of the numbers 1 2 3 ... n, as 
long as (1) (bf h) )(c^) has no two numbers alike; and (2) no numbers 
in (Jbf h) ) and (n | m | k ) are alike. 
a p +t 
h 
