818 N U M 
its horizontal rows, the natural, triangular, pyramidal, 
Sec. numbers ; for, in the fecond, we have the natural 
numbers, i, z, 3, 4, See. in the third, the triangular num¬ 
bers, 1, 3, 6, 10, 15, &c. in the fourth, the pyramidals of 
the firfl order, 1, 4, 10, 20, 35, See. in the fifth, the pyra- 
midals of thefecond order, 1, 5, 15, 33, 70, See. This is a 
neceflary confequence of the manner in which the Table 
is formed ; for it may be readily perceived, that the num¬ 
ber in each fquare is always the fum of thofe which fill the 
preceding fquares on the left, in the row immediately 
above. The tame numbers will be found in the rows pa¬ 
rallel te> the diagonal, or the hypothenufe of the triangle. 
But a property flill more remarkable, which can be com¬ 
prehended only by fuch of our readers as are acquainted 
with algebra, is, that the perpendicular rows exhibit the 
co-efficients belonging to the different members of any 
power to which a binomial, as a-\-b, can be raifed. The 
third row contains thofe of the three members of the 
fquare; the fourth, thofe of the four members of the 
cube; the fifth, thofe of the five members of the biqua¬ 
drate. 
By combinations, are underftood the various ways that 
different things, the number of which is known, can be 
cholen orfeledled, taking them one by one, two by two, 
three by three, Sec. without regard to their order. Thus, 
for example, if it were required to know in how many dif¬ 
ferent ways the four letters a, b, c, d, could be arranged, 
taking them two and two, it may be readily feen that we 
can form with them the following combinations, ab, ac, 
od, be, bd, cd; four things, therefore, may be combined, 
two and two, fix different ways. Three of thefe letters 
may be combined four ways, nbc, abd, acd, bed; hence 
the combinations of four things, taken three and three, 
are only four. 
In combinations properly fo called, no attention is 
paid to the order of the things ; and for this reafon we 
have made no mention of the following combinations, 
ba, ca, da, cb, db, dc. If, for example, four tickets, marked 
a, b, c, d, were put into a hat, and any one fhould bet to 
draw out the tickets a and d, either by taking two at one 
time, or taking one after another, it would be of no im¬ 
portance whether a fhould be drawn firfl or lafl; the com¬ 
binations cd or da ought therefore to be here confidered 
only as one. 
But, if any one fhould bet to draw out a the firfl time, 
and d the fecond, the cafe would be very different; and 
it would be neceflary to attend to the order in which 
thefe four letters may be taken and arranged together 
two and two; it may be eafily feen, that the different ways 
are 12 ; ab, ba, ac, ca, ad, da, be, cb, bd, db, cd, dc. In like 
manner, thefe four letters might be combined and ar¬ 
ranged, three and three, 24 ways, as abc, acb, bac, bca, 
cab, eba, adb, abd, dba, dab, bad, bda, acd, adc, dac, dca, cad, 
eda, bed, die, eld, bde, cbd, deb. This is what is called per¬ 
mutation, and change of order. 
Merfenne gives us the combination of all the notes and 
founds in mufic, as far as fixty-four ; the fum whereof 
amounts to ninety figures, or places. 
The number of poffible combinations of the twenty- 
four letters of the alphabet, taken firfl two by two, then 
three by three. See. according to Mr. Preftet’s calculation, 
amounts to 1391724288887252999425128493402200. 
Prob. I. Any number of things whatever being given; to 
determine in how many ways they may be combined two and 
two, three and three, fyc. without regard to order. —This 
problem may be eafily folved by making life of the arith¬ 
metical triangle. Thus, if there are eight things to be 
combined three and three, we mult take the ninth vertical 
row, or in all cales that row the order of which is expreffed 
by a number exceeding by unity the number of things to 
be'combined ; then the fourth horizontal row, or that the 
order of which is greater by unity than the number of 
the things to be taken together; and in the common fquare 
of both will be found the number of the combinations 
required, which, in the prefent example, will be 56. 
BEK. 
But, as an arithmetical triangle may not always be at 
hand, or as the number of things to be combined may be 
too great to be found in fuch a table, the following fimpie 
method may be employed. The number of the things to 
be combined, and the manner in which they are to be 
taken, viz. two and two, or three and three. Sec. being 
given : . Form two arithmetical progrelfions, one in which 
the terms go on decreafing by unity, beginning with the 
given number of things to be combined; and the other 
confining of the feries of the natural numbers, 1,2, 3, 4, 
Sec. Then take from each as many terms as there are 
things to be arranged together in the propofed combina¬ 
tion. Multiply together the terms of the firfl progreffion, 
and do the fame with thofe of tiie fecond. In the lafl 
place, divide the firfl product by the fecond, and the quo¬ 
tient will be the number of the combinations required. 
Ex. 1. In how many ways can 90 things be combined , 
taking them two and twol —According to the above rule 
we mufl multiply 90 by 89, and divide the produdl, 8010, 
by the produdl of 1 and 2, that is, 2 ; the quotient, 4005, 
will be the number of the combinations refulting from 
90 things taken two and two. 
Should it be required, in hoiv many ways the fame 
things can be combined three and three, the problem may 
be anfwered with equal eafe ; for we have only to multi¬ 
ply together 90, 89, 88, and to divide the produdl, 704880, 
by that of the three numbers 1,2, 3 ; the quotient, 117480, 
will be the number required. In like manner, it will be 
found, that 90 things rhay be combined, by four and four, 
2 555t9° ways; for, if the produdl of 90, 89, 88, and 87, be 
divided by 24, the produdl of 1, 2, 3, 4, we fhall have the 
above refillt. In the lafl place, if it be required, what 
number of combinations the fame 90 things, taken five 
and five, are fufceptible of, it will be found, by following 
the rule, that the anlwer is 43949268. 
Ex. 2. Wereitafked, how many conjundlions the feven 
planets could form with each other, two and two, we might 
reply 21 ; for, according to the general rule, if we multi¬ 
ply 7 by 6, which will give 42, and divide that number by 
the produdl of 1 and 2, that is, 2, the quotient will be 21. 
If we wifhed to know the number of all the conjunc¬ 
tions poffible of thefe feven planets, two and two, three 
and three. Sec. by finding feparately the number of the 
conjundlions two and two, then thofe of three and three, 
Sec. and adding them together, it will be feen that they 
amount to 120. The fame relult might be obtained, by 
adding the feven terms of the double geometrical pro¬ 
greffion, 1, 2,4, 8, 16, 32, 64, which will give 127. But 
from this number we mufl dedudl 7, becaufe, when we 
fpeak of the conjundlions of a planet, it is evident that two 
of them, at leafl, mull be united; and the number 127 
comprehends all the ways in which leven things can be 
taken, one and one, two and two, three and three, &c. In 
the prefent example, therefore, w r e mufl dedudl the num¬ 
ber of the things taken one and one ; for a fingle planet 
cannot form a conjundlion. 
Prob. II. Any number of things being given; to find in 
how many ways they can be arranged. —This problem may 
be eafily folved, by following the method of indudlion ; 
for, 1 ft. One thing, a, can be arVanged only in one way; 
in this cafe, therefore, the number of arrangements is — i. 
2d. Two things maybe arranged together two ways; for, 
with the letters a and b, we can form the arrangements ab 
and ba ; the number of arrangements, therefore, is equal 
to 2, or the produdl of 1 and 2. 3d. The arrangements 
of three things, a, b, c, are in number fix ; for, ab can form 
with c, the third, three different ones, bac, bca, eba, and 
there can be no more. Hence it is evident, that the re¬ 
quired number is equal to the preceding multiplied by 3, 
or to the produdl of 1, 2, 3. 4th. If we add a fourth thing, 
for inftance, d, it is evident that, as each of the preceding 
arrangements may be combined with this fourth tiling 
four ways, the above number, 6, mufl be multiplied by 4, to 
obtain that of the arrangements refulting from fourthings ; 
that is to fay, the number will be 24, or the produdl of 
3 » +* 
