170 MAJOK P. A. MAcMAHON: MEMOIR ON THE 



is an arrangement enumerated by (53 ;), I take the numbers 8, 7, 6, 5, 4, 3, 2, 1 in 

 descending order and write down a when the number is in the first row and /8 when 

 it is in the second, thus 



87654321 



and we now have an arrangement of the letters in 



6 /8 3 



such that, proceeding from left to right and stopping at any point, the number of a's 

 met with is at least as large as the number of yS's. 



The result and correspondence are at once generalisable, for consider the arrangement 



9765 



832 



4 



1 ; 



we are led to the permutation of 4 y6 3 yS, viz., 



which is such that, in passing from left to right, we have at any instant 



(i) passed over at least as many a's as /3's, 



(ii) ,, 's y's, 



(iii) y's S's. 



Hence, if in an election four candidates have a, b, c, d (these numbers being in 

 descending order of magnitude) supporters respectively, and at any instant they have 

 respectively A, B, C, D votes, the probability that always 



s 



and, in general, if the votes polled at any instant show invariably the final order of the 

 candidates, we have a state of affairs of which the probability is 



23. From the result 



(abode... ;) - ( + b + c + d+e + ...)\ 

 V alblc\dle\... 



