340 
Proceedings of the Boyal Society 
Next for two conditions ; these may be [2], say the conditions 
are 1 and 2, or else [1 , 1] say they are 1 and 3. In the former 
case the arrangements which fail are those which contain in the 
first and second places ab, ac , or be, and for each of these the other 
n — 2 letters may be arranged in any order whatever; there are 
thus 3 II (n- 2) failing arrangements. In the latter case the 
failing arrangements have in the first place a. or b, and in the third 
place c or d , — viz., the letters in these two places are a. c, a.d, 
b.c, or b.d, and in each case the other n- 2 letters may be 
arranged in any order whatever : the number of failing arrange- 
ments is thus = 2 . 2 . II (n - 2) . And so in general when the con- 
ditions are [a, p, y . .], the number of failing arrangements is 
= ('0,4- l)(p + l)(y + 1) . . n(n - a - j3 - y . .) . But for [n], that is for 
the entire system of the n conditions, the number of failing 
arrangements is (not as by the rule it should be = n + 1 , but) = 2, 
— viz., the only arrangements which fail in regard to each of the 
n conditions are (as is at once seen), abc . .jk, and be. ..jka. 
Changing now the notation so that [1], [2], [1,1], &c., shall 
denote the number of the conditions [1], [2], [1,1], &c., respec- 
tively, it is easy to see the form of the general result : if, for 
greater clearness, we write n = 6, we have 
1 -S(l) +2(12) -2(123) 
No. = 720 - {([1] = 6)2} 120 + / ([2] =6)3 124- 
( ((3] 
= 6)4 \ 
\ + ([1,1] = 9)2.2/ 
+ ([2,1] 
= 12)3.2 ( 
1 j / | 
( + ([ 1 , 1 , 1 ]= 2 . 2 . 2 ) 
+ 2(1234) -2(12345) +(123456) 
+ ( ([4 )-6)5 n2 - {([5] = 6)6} 1 +{([6] = 1)2} 
] +([3,1] = 6)4.2 1 
(+([2,2] = 3)3.3) 
or, reducing into numbers, this is 
No. = 720 - 1440 + 1296 - 672 + 210 - 36 + 2 , = 80 . 
The formula for the next succeeding case, n= 7, gives 
No. = 5040 - 10080 + 9240 - 5040 + 1764 - 392 + 49 - 2 , = 579 . 
