669] 
245 
669 
ON A PROBLEM OF ARRANGEMENTS. 
[From the Proceedings of the Royal Society of Edinburgh, t. ix. (1878), pp. 338—342.] 
It is a well-known problem to find for n letters the number of the arrangements 
in which no letter occupies its original place; and the solution of it is given by 
the following general theorem:—viz., the number of the arrangements which satisfy 
any r conditions is 
(1-1) (1-2) (1 - r), 
= 1 -2 (l) + 2 (12)- ±(12...r), 
where 1 denotes the whole number of arrangements; (1) the number of them which 
fail in regard to the first condition; (2) the number which fail in regard to the 
second condition; (12) the number which fail in regard to the first condition, and 
also in regard to the second condition; and so on: 2(1) means (1) + (2) + ... + (r): 
2(12) means (12) + (13) + (2r) + ... + (r — 1, r); and so on, up to (12...r), which denotes 
the number failing in regard to each of the r conditions. 
Thus, in the special problem, the first condition is that the letter in the first 
place shall not be a; the second condition is that the letter in the second place 
shall not be b; and so on; taking r = n, we have the known result, 
n.n— 1...2.1 
1.2...n 
No. = lift — j IT(w — + II (w — 2). + ... + 
giving for the several cases 
n = 2,3,4, 5, 6, 7,... 
No. = 1, 2, 9, 44, 265, 1854,... 
I proceed to consider the following problem, suggested to me by Professor Tait, 
in connexion with his theory of knots: to find the number of the arrangements of 
n letters abc... jk, when the letter in the first place is not a or b, the letter in 
the second place not b or c,, the letter in the last place not k or a. 
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