360 



PKOFESSOR KELLAND ON A 



By placing A and B showing face 1, and C and D showing face 2, with each of 

 these, you have the arrangements in which two particular duplications of the dice 

 A, B, C, D, and those only, occur. Now the number of arrangements of faces 1 

 and 2, on dice A, B, C, D, is the number of permutations, all together, of four 



4.3.2.1 



things, two of one kind and two of another, or, 

 4.3.2.1 



Hence 



(p-2) (p-3) 



1" . 2 2 • 



. (p — n + 3) 



is the number of arrangements in which two duplications occur of faces 1 and 2, 

 and on dice A, B, C, D. The same is true of any other pair of faces ; consequently 

 the number of arrangements in which two duplications are found on the dice A, 

 B, C, D, but on no others, nor any repetition of the faces shown on these four 

 dice, is — 



P -^ • ^A (P-2) (P-3) .... ( P -„ + 3). 



In like manner, any other four dice form the same number of arrangements ; and 

 hence the total number of arrangements in which two duplications and no more 

 occur, is — 



4.3.2.1 n(n -l) (n-2) (n-3; 1 . ^ . _. 



— w~ * — 1.2,3.4 — • X7g P (p-i) -■ • • • (P~ n + S )- 



4. Similarly, the number of arrangements in which three duplications, and 

 three only, occur without any other repetition, is, — 

 6.5.4.3.2.1 n(n-l). 



" ' 17273 ^-P -1 ) 



(p-n + 4), 



2 3 1.2 .... 6 



and the law of formation is evident. 



5. We may now write the number of arrangements in which no triplication 

 occurs, in the following form : — 



(p-n + 1) + ^-y p (p-1) .... {p-n + 2) 



p(p-l) . . 



4 . 3 . 2 . 1 , n (n-1) (n-3) 



6.5 



.1 



1.2.3.4 

 n (n — 1) .... (n — 5) 



1 

 1.2 



P O- 1 ) 



1.2 



6 



i7273^ (i?_1) 



/ i\ / -in f -i n (n — 1) 1 



=p 0-1) • • • • (p-n + 1) j 1 + 



(p—n + 3) 

 . (p-n + 4:) + &c. 



1 

 1.2 



F2 + 



n (n — 1) 



p — n + 1 



(n-5) 



+ 



T^H-to. } 



n (n — 1) .... (n — 3) 



(p-n + 1) (p-n + 2) ' 1.2 ' 2 2 ' (p-n + 1) (p-n + 2) (p-n+ 3) 1.2.3 



6. This series may be exhibited as the solution of a differential equation, but 

 it is doubtful whether, with our present knowledge, we can simplify its form. We 

 obtain the differential equation thus : — 



Let 



m = 1+ njn-l) x] n(n-l) . . . (n-3) ^ 



p — n + 1 (p — n + 1) (p — n + 2) ' 1 



— o+ &c. 



