362 PROFESSOR KELLAND ON A PROBLEM IN COMBINATIONS. 
is the number in which A, B, C, D, E, F form two triplets. Hence the number 
of arrangements in which there are two triplets, and no more, is— 
n(n—1)....(@m—5) 6.5... .1p (p— ) —6 
1a eee (1.2.3)? 1.2 Or-2 
SRG Se=h Pe) gs 
(1.2.3) 
9. The whole number of arrangements in which no quadruplication occurs is— 
C, 7 ee Pp bea tee le Cos 2 3 . (n—5) p — 1) ca 
n(n—1).. . .(n—8) p (p—1) (p—2) -n 
2 3) Ee Sat be. 
10. In the same manner it may be shown, that, if the above series be repre- 
sented by D’, the whole number of arrangements in which no quintuplications 
occur is— 
nn (n—1) (n—2) (n—3) n—4 n(n—1)....(n—7) , p (p—1) pn-8 
es 1.2.3.4 PD sat ~ 7.9.8.4 io Op aes 
11. It is evident that the total number of arrangements of the faces is p”. 
Hence the probability that no two show the same face is— 
P(p—)).. - -(p=n+I). 
7 
