.TRANSACTIONS OP THE SECTIONS. 5 



or 



l + (a + b + c + d)Px + {ab + ac + ad+bc+bd+cd)P^x^ 

 + (abc+abd+acd+ bcd)F^x^ + aicrfPV. 



The number of the permutations in each case is given by the coefficients 

 of the several powers of x in the expansion of 



( 1 + P^)' or 1 + 4Pa: + 6P V + 4P V + P V. 

 That is, 



4Pj = 4P=4: 4P^=6P2=12: 4P3=4P3=24 : 4P4=P*=24. 

 Next consider the case of a, a, c, d. 



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

 the expansion of 



fl + a.'Px^- -^.PvVl + c.PajXl + rf.Pa;), 



or 



\ + {a + c+d)'Px+(ac-\-ad+cd+ ^—\V^3^ 



+ f^!lI±^+«crf)pV+ tSl PV. 

 V 1.2 / 1.2 



The justice of this conclusion may be seen by examining the mode of for- 

 mation of each coefficient. 



The number is found by equating a, c, d to unity : 



1 + 3Pa; + |PV + 2PV + iPV. 

 Hence 



4P, = 3:4P2=7:4P3=12:4P4=I2. 



The general theorem may be expressed as follows : — 



nPj.= the coefficient of a;* in the expansion of 



I 



V 1 . 2 W J 



r, ,-D ^ pv^ , (P^)?\ 

 x(^l + P..+ — + ..+L_Lj 



x(^l + Pa.+ _+..+L_^j 



X 



where P is subject to the law of indices. 

 We may observe that 



pra ■ n 



«P»= 



a well-known theorem. 



6. The total number of permutations of n things taken 1, 2, 3 

 together is 



