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) . . . . (n — 5) 6.5. . . .lp(p — 1) p«-e 



1 .2. . . 6 



n (n— 1) . 



(1 . 2 . 3) 2 1.2 



♦ • (w-5 ) p Q-l) r "- 6 . 

 (1 . 2 . 3) 2 1.2 *- 2 



9. The whole number of arrangements in which no quadruplication occurs is- 



C, 



" + »(»-!) 0*~ 2 ) q"- 3 + w(n-l). • ■ • (n-5 ) j>(j?-l) c »-« 



1.2.3 



p-i 



1.2 



p-3 



+ 



(1.2. 3) 2 



n (n-1) . . . . (n-8) j> (p-1) (g-2) r »-» . 

 (1.2.3) 3 1.2.3 %-3 + <KC - 



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 — 



» n(n-l) (n-2) (n-3) n *-4 n (n-1) . . . . (n-7) 

 ^ p+ 1.2.37T" ^^-1+ n • 2 . 3 . 4) 2 



1} D""' + &c. 

 f>— s 



(1 . 2 . 3 . 4) 2 1.2 



11. It is evident that the total number of arrangements of the faces is jo". 

 Hence the probability that no two show the same face is — 



p (p— 1) . . . .(p—n + 1). 



