382 
Proceedings of the Royal Society 
5. On Professor Tait’s Problem of Arrangement. By 
Thomas Muir, M.A. 
The problem in question is — To find the number of possible 
arrangements of a set of n things, subject to the conditions that 
the first is not to be in the last or first place, the second not in the 
first or second place, the third not in the second or third place, 
and so on. 
A little consideration serves to show that we may with advan- 
tage shift the ground of the problem to the theory of determinants. 
For the sake of definiteness take the case of five things, A, B, C, 
D, E. Here A may be in the second, third, or fourth places only ; 
B in the third, fourth, or fifth places only ; and similarly of the 
others — a result which we may tabulate thus : — 
AAA. 
. . B B B 
c . . c c 
D D . . J) 
E E E . . 
an A being written in the places which it is possible for A to 
occupy, and a dot signifying that the letter found in the same 
line with it may not occupy its place. Hence, to obtain the 
various arrangements, we see that for the first place we may have 
any letter that is in the first column ; for the second place any 
letter that is in the second column, provided it be not in the same 
line with the letter taken from the first column ; for the third 
place, any letter that is in the third column, provided it be not in 
the same line with either of the letters previously taken, and so 
on. This law of formation, however, is identical with that in 
accordance with which the terms of a determinant are got from 
the elements of the matrix ; so that the problem we are concerned 
with is transformed into this : Find the number of terms of the 
determinant of the wth order of the form, — 
