NUMBER. 310 
t, 3, 3, 4. It is needlefs to enlarge farther on this fubjeft; 
for it may be eafily feen that, whatever be the number of 
the things given, the number of the arrangements they are 
fufceptible of may be found, by multiplying together as 
many terms of the natural arithmetical progreflion as there 
are things propofed. This is the bafis of the large Table 
we (hall infert prefently. 
It may fometimes happen that, of the things propofed, 
one of them is repeated, as a, a, b,c. In this cafe, where 
two of the four things propofed are the fame, it will be 
found, that they are fufceptible only of 12 arrangements, 
inftead of 24; and that five, where two are the fame, can 
form only 60, inltead of 120. But, if three of four things 
were the fame, there would be only 4 combinations inftead 
of 245 and five things, if three of .them were the fame, 
would give only 20, inftead of 120, or a fixth part. But, 
as the arrangements of which two things are fufceptible 
amount to 2, and as thofe which can be formed with 
three things are 6, we may thence deduce the following 
Rule : In any number of things, of which the different 
arrangements are required, if one of them be feveral times 
repeated, divide the number of arrangements, found ac¬ 
cording to the general rule, by that of the arrangements 
which would be given by the things repeated, if they were 
different, and the quotient will be the number required. 
Again, in the number of things, the different arrange¬ 
ments of which are required, if there are feveral of them 
which occur feveral times, one twice for example, and an¬ 
other three times; nothing will be neceflary, but to find 
the number of the arrangements, according to the general 
rule, and then to divide it by the product of the numbers 
expreffing the arrangements which each of the things-re¬ 
peated would be fulceptible of, if, inftead of being the- 
fame, they were different. Thus, in the prefent cafe, as 
the things which occur twice would be fufceptible of two 
arrangements, if they were different; and as thofe which 
occur thrice would, under the like circumllances, give 
fix ; we mult multiply 6 by 2, and the produft, 12, will be 
the number, by which that, found according to the general 
rule, ought to be divided. Thus, for example, the five 
letters a, a, b, b, b, can bearranged only 10 different ways ; 
for, if they were different, they would give 120 arrange¬ 
ments ; but, as one of them occurs twice, and another 
thrice, 120 mull be divided by the product of 2 and 6, or 
12, which will give 10. 
By obferving the precepts given for the folution of this 
problem, the following queftions may be refolved. 
1. A club of J'even perfons agreed to dine together, evert/ 
day fuccejfively, as long as they could Jit down to table diffe¬ 
rently arranged. How many dinners would be neceffary for 
that purpoj'e? —It may be eafily found, that the required 
number is 5040, which would require 13 years and more 
than 9 months. 
2. The different anagrams which can be formed with 
any word, may be found in like manner. Thus, for ex¬ 
ample, if it be required, how many different words can be 
formed with the four letters of the word AMOR, which 
will give all the poftible anagrams of it, we fhall find that 
they amount to 24, or the continued product of 1, a, 3, 4. 
We fhall here give 
them in their regular 
order: 
AMOR 
MORA 
ORAM 
RAMO 
AMRO 
MOAR 
ORMA « 
RAOM 
AOMR 
MROA 
OARM 
RM AO 
AORM 
MRAO 
OAMR 
RMOA 
ARMO 
MAOR 
OMR A 
ROAM 
AROM 
MARO 
OMAR 
ROMA 
Hence it appears, that the Latin anagrams of the word 
amor, are in number feven, viz. Roma, mora, maro, oram, 
ramo, anno, orma. But, if in the propofed word, one or 
more letters were repeated, it would be neceflary to fol¬ 
low the preceptsalready given. Thus, the word Leopoldus, 
where the letter l and the letter o both occur twice, is fuf¬ 
ceptible of only 90720 different arrangements, or ana¬ 
grams, inftead of 3628800, which it vfould form if none 
of the letters were repeated ; for, according to the before- 
mentioned rule, we niuft divide this number by the pro¬ 
duct of 2 by 2, or 4, which will give 90720. The word 
JludioJ'us, where the u occurs twice and the s thrice, is 
fufceptible of only 30240 arrangements; for, the arrange¬ 
ments of the 9 letters it contains, which are in number 
3628800, mult be divided by the produft of 2 and 6, or 
twelve, and the quotient will be 30240. 
In this manner may be found the number of all the 
poftible anagrams of any word whatever; but it mull be 
oblerved, that, however few be the letters of which a word 
is compofed, the number of the arrangements thence re- 
fulting will be fo great as to require confiderable labour 
to find them. Hence the utility of a Table. 
3. How many ways can the following verje be varied, with¬ 
out deftroying the meaj'ure: “ Tot tibi J'unt dotes, Virgo, 
quot Jidera ccelo?" —This verfe, the production of a devout 
Jefuit of Louvaine, named father Bauhuys, is celebrated 
on account of the great number ofarrangements of which 
it is fufceptible, without the laws of quantity being vio¬ 
lated ; and various mathematicians have exerciled or 
amufed themfelves with findingout thenumber. Erycius 
Puteanus took the trouble to give an enumeration of them 
in forty-eight pages, making them amount to 1022, or the 
number of the ltars comprehended in the catalogues of 
the ancient aftronomers ; and he very devoutly obierves, 
that the arrangements of thefe words as much exceed the 
above number as “ the perfeCfions of the Virgin exceed 
that of the ftars.” Father Preftet, in the firft edition of 
his Elements of the Mathematics, fays, that this verfe is 
fufceptible of 2196 variations ; but, in the 1'econd edition, 
he extends-the number to 3276. Dr. Wallis, in the edition 
of his Algebra printed at Oxford in 1693, makes them 
amount to 3096. But none of them has exadily hit the 
truth, as has been remarked by James Bernoulli, in his 
Ars Conjcftandi. This author lays, that the different 
combinations of the above verfe, leaving out the fpondees, 
and admitting thofe which have no csefura, amount ex¬ 
actly to 3312. The method by which the enumeration 
was made, may be feen in the above work. 
The fame queftion has been propofed refpefting the fol¬ 
lowing verfe of Thomas Lanfius: Mars, mors, fors, lis, 
vis,jlyx, pus, nox,fex, mala, crux, fraus. It may be eafily 
found, retaining the word mala in the autepenultima 
place, in order to preferve the meafure, that this verfe is 
fulceptible of 39916800 different arrangements. 
Prob. III. Of the combinations which may be formed with 
fquares divided by a diagonal into two differently-coloured 
triangles. —We are told by father Sebaltian Truchet, of 
the Royal Academy of Sciences, in a memoir printed 
among thofe of the year 1704, that having feen, during 
the courfe of a tour which he made to the canal of Orleans, 
fome fquare porcelain tiles, divided by a diagonal into 
two triangles of different colours, deftined for paving a 
chapel and fome apartments, he was induced to try in 
how many different ways they could be joined fide by fide, 
in order to form different figures. In the firft place, it 
may be readily feen that a fingle fquare, according to its 
pofition, can form four different figures; which, however, 
may be reduced to two, as there is no other difference be¬ 
tween the firft and the third, or between the fecond and 
the fourth, than what arifes from the tranfpofition of the 
fliaded triangle into the place of the white one. 
Now, if two of thefe fquares be combined together, 
the refult will be 64 different ways of arrangement; for, 
in that of two fquares, one of them may be made to al- 
fume four different lituations, in each of which the other 
may be changed 16 times. The refult, therefore, will be 
64 combinations. We muft however oblerve, with father 
Sebaftian himfelf, that one half of thele combinations 
are only a repetition of the other in a contrary direction, 
which reduces them to 32 ; and, if attention were not 
paid to fituation, they might be reduced to 10. 
4 
