4 REPORT 1856. 



Hence we conclude that there is no necessity for investigating the coeffi- 

 cients of powers of x beyond — if » be even, or beyond if n be odd. 



This consideration vastly diminishes the labour of expansion. 



3. The total number of possible combinations is found by equating x to 

 unity in the formula (A), and subtracting 1 from the result, since 1 is the 

 first term in the expansion involving no power of x, and therefore cannot 

 denote the number of any combination. 



Hence the number required is 



{p.+ l){q+\){r+\) -1; 



which is a known theorem. 



4. Example : To find the number of combinations that can be formed of 

 the letters of the word "Notation" taken severally 1, 2, 3, ... 8 together. 



There are two w's, two o's, two ^'s, one a, one i. 



The numbers required are found by expanding, at least as far as x*, 



(1 -a^f^\-xy ^ ^^ _^y^i_^y. . (l - ^■) -^ 



= (l-3;r'+ ){\-23? + x^) 



X (1 +5x+\53^-\-Z5x^+ 10x*+. . ) 



= l+5x+lSx^+22x^+26x^+.. .. 



The series can now be completed by aid of the theorem 



^8-4 = 8^*: 



l+5a:+13r^ + 22,r3 + 26^H22a;'+13^H5^^ + ;r«. 

 The total number of possible combinations 



=5 + 13 + 22 + 26 + 22 + 13 + 5 + 1 = 107 = 3.3.3.2.2-1. 

 as might have been obtained at once by the formula 

 (p+l)(q+l)(r+l) ... -I. 



This example was selected to contrast the tentative method used in ' Lund's 

 Companion to Wood's Algebra,' p. Ill, London, 1852, in the particular case 

 of k=3. 



I quote the author's words : — 



" Here are five different letters : the number of combinations of five letters, 



5x4 



3 together, where no letter recurs = =10. 



^ 1x2 



" Also there are two «'s, two o's, and two <'s, each of which pairs may be 



combined with each of the other four letters, and form four combinations of 



three, making altogether 3 x 4=; 12 such combinations where the letters recur, 



•.• number required =10+ 12=:22." 



5. To find the number of permutations of n things taken 1, 2, 3, . . « 

 together, when n consists of groups of identical quantities, p of one sort, q of 

 another, r of another, &c. 



In the following solution we shall denote ^Pj, 2P2. • • r^r by powers of P, 

 viz. P, P-, ... P'', and subject P to the laios of indices. 



In order to see more clearly the method and notation that will be adopted, 

 let us examine the familiar case of four different quantities, a, b, c, d. The 

 permutations are contained in the coefficients of the several powers of x in 

 the expansion of 



(1 + Pa,r)(l + P6x)(l + Pcci-)(1 + Prfx), 



