360 PROFESSOR 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 
things, two of one kind and two of another, or, Saige 
3.2.1 
Hence = = es) Pe 3)... .(p—n+8) 
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, ©, D, but on no others, nor any repetition of the faces shown on these four 
dice, is— 
pie). 4.3. oe i 
(p= 2) Gm vn cpa 49): 
In like manner, any ie 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—1) (n—2) (n—3) 1 
Fatal . ya 1. 7-9 P (p—1)-- - -(p—n+8). 
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 nm) %.: m—5) 1 
28 oo ‘To7g P P-D)--- -(w—nt4), 
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(p—-1)..--(p—nt) ye a p (p—1)... . (p—n+2) 
f eet p not) 28 . ae pel oLion Ge Saw 
4 Oe Oe Teer). pond) 
=p (p-1)--.-(p—ntt) { 14525). 54 
n(n—1)....(@—3) 1 1s n (m—1)... .(n—5) i re 
(p— Ss aee —n42)' 1.2° 21 (p—n+l) (p— ae n+ 6) 1.2.3° zt ie } 
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 :-— 
_,, n(n—1) nm (n—1). (n—3) a 
Let oot eae (p—n-+1) (ee m+2) * G2 at 
