102 
Proceedings of Boyal Society of Edinburgh. 
Note on Selected Combinations. By Thomas Muir, LL.D. 
(Read January 6, 1902.) 
(1) Having formed from n things all possible sets of r , we may 
subject each of the C n>r sets to the test of fulfilling one or more 
conditions, and so obtain a reduced number possessing a special 
characterisation. The present note deals with a few instances of 
this in which the reduced number is of the same algebraical form 
as the full number,* that is to say, is a combinatorial C a>& . 
(2) First let us take the condition that the set must not contain 
two things which in the original row of n things occupied con- 
secutive places, and let us denote by C n>r the reduced number due 
to the imposition of this condition. 
Denoting the things by 
«i, a 3> • • • j CL n 
we see that for r = 2 all the possible combinations are got by 
taking 
(1) aj along with a 3 , a 4 , a b , ... , a n 
(2) a 2 a 4 , a 5 , . . . , a n 
(3) a s a 5 , . . . , a n 
and that therefore 
C Wj 2 = {n — 2) + (n — 3) 4- ... +1, 
’ =i( w -l)(*-2), 
= CU, 2 . 
The general theorem which we propose to establish, viz. 
C n,r r+1, r 
is thus seen to be true for the case where r = 2. But if it be true 
for any particular value of r , say r = s , it can be shown to be 
* An analogous peculiarity appears in the well-known theorem that the 
number of combinations of n things taken r at a time when repetitions are 
allowed is the same as the number of combinations of n-r+ 1 things with 
repetitions debarred, i.e ., is equal to C n _r+i,r» 
