8o Dr. Young’s remarks on the reduction 
S • • n; and that of these the number in which no one name 
agreed would be a = m — a — n . . a — n . . 
. . — n . a n __ x ; each term expressing the number of agree- 
ments in n, n — i, n — 2 . . . instances only, and being made 
up of all the combinations of so many out of n things, each 
occurring as many times as all the remaining ones can dis- 
agree. Hence we may easily obtain the successive values of 
a from each other, the first being obviously l, as a single 
name can only be given in one way to a single thing, there- 
fore, 
a 2 — 2 — I = I 
<*3 = 6 — 1—3 = 2 
a ^ ~ 24— 1 — 6 — 8 := 9 
a . — 120 — 1— 10—20 — 45 — 44 
— 720 — 1 — 15 — 40— 135 — 264 — 265 
a-j “ 5040 — 1 — 21 — 70 — 315 — 924 — 1855 — 1854 
a g =: 40320 — 1—28 — 1 12 — 630 — 2464 — 7420 — 14832 ~ 14833 
« 9 = 3 6z88 ° — 1 — 36 — 168 — 1 134 — 5544 — 22260 — 66744 — 133497 = 133496 
a lQ — 3628800—1 — 45 — 240 — 1890 — 11088 — 55650 — 222480 — 667485 — 1334960 
= I3349 61 
From this computation it may be inferred, that, for 10 
names, the probabilities will stand thus : 
No coincidence 
.367880 
One or 
more 
.632120 
One only 
.367880 
Two or 
more 
.264240 
Two only 
.18394,1 
Three or more 
.080300 
Three only 
.061309 
Four or 
more 
.018991 
Four only 
15336 
Five or 
more 
.003655 
Five only 
.003036 
Six or more 
.000599 
Six only 
.000321 
Seven or more 
.000078 
