COM 
dies by fire, does not fo much enrich mankind as it di¬ 
vides the bodies ; as upon the fcoreof its making new 
compounds by new combinations. Boyle .—Ingratitude is 
always jn combination with pride and hard-heartednefs. 
South .—Copulation of ideas in the mind —They never 
i'uffer any ideas to be joined in their underftandings, in 
any other or ftronger combination than what their own 
nature and correfpondence give them. Locke. — Combina¬ 
tion is ufed, in mathematics, to denote the variation or 
alteration of any number of quantities, letters, founds, 
See. in all the different manners poffible. Thus the num¬ 
ber of poflible changes or combinations of the twenty-four 
letters of the alphabet, taken firft two by two, then three 
by three, and lb on, according to Preftet’s calculation, 
amount to 1,391,724,288,887,252,999,425,128,493,402,200. 
Father Merfenne gives the combinations of all the 
notes and founds in mulic, as far as 64; the fum of which 
amounts to a number exprelfed by 90 places of figures. 
And father Trucliet, in Mem. de l’Acad. fhews, that two 
fquare pieces, each divided diagonally into two colours, 
may be arranged and combined 64 different ways, fo as 
to form fo many different kinds of chequer-work; a thing 
that may be of ufe to mafons, paviours, painters, Sec. 
Doctrine of Combinations.— 1. Having given any 
number of things, with the number in each combination; 
to find the number of combinations. When only two 
are combined together. 
One thing admits of no combination. 
Two, a and b, admit of 1 only, viz. ab. 
Three, a, b, c, admit of 3, viz. ab, ac, be. 
Four admit of 6, viz. ab, ac, ad, be, bd, cd. 
Five admit of 10, viz. ab, ac, ad, ae, be, bd, be, cd, ce, de. 
Whence it appears that the numbers of combinations, of 
two and two only, proceed according to the triangular 
numbers 1, 3, 6, 10, 15, 21, &c. which are produced by 
the continual addition of the ordinal feries o, 1, 2, 3, 4, 5, 
Sec. And if n be the number of things, then the general 
formula for expreffing the fum of all their combinations 
, , n.n —1 
by twos, will be-■. 
1.2 
^ • I 
Thus, if n — 2 ; this becomes --~ 1, 
2. 
ic .. . 3 • * 
If n — 3; it is - - - =2 3. 
2 
If ft — 4; it is J —— - - =6. Sec. 
Z 
When three are combined together 5 then 
Th ree things admit of 1 order, abc. 
Four admit of 4 5 viz. abc, abd, acd, bed. 
Five admit of 10 ; viz. abc, abd, abe, acd, ace, ade, bed, 
bee, bde, cde. And fo on according to the firft pyramidal 
numbers 1,4, 10, 20, Sec. which are' formed by the conti¬ 
nual addition of the former, or triangular numbers i, 3, 
6, 10, Sec. And the general formula for any number n of 
, . . , , , . n . v — 1 . h —• 2 
combinations, taken by threes, is —-, 
1.2.3 
. . . 3.2.1 
So, if n — 3 j it is --— 1. 
1.2.3 
-- .. 4 - 3 .Z 
It n — 4; it is - — 4. 
6 
If n — 5; it is ————- — 6* ^ c - 
Proceeding thus, it is found that a general formula for 
any number n of things, combined by m at each time, 
n.n — 1 . n — 2 . n — 3. &c . . . 
is s— -rr-, continued to m fac- 
1 . 2 . 3 . 4. Sec 
tors, or terms, or till the laft faftor in the denominator 
be m. So, in 6 things, combined by fours, the number of 
, - .. ■ 6 • 5 • 4 • 3 _ 
combinations is-- — t 5. 
1 . 2 . 3 . 4 
By adding all thefe feries together, their fum will be the 
whole number of poffible combinations of n things com- 
Vol, IV. No. 243. 
COM 845 
bined both by twos, by threes, by fours, Sec. And as the 
laid feries are evidently the coefficients of the power n of a 
binomial, wanting o"nly the firft two 1 and n\ therefore the 
lai d fum, or whole number of all fuch combinations, will 
be 1 -)- iT — n — 1, or x n — n — 1. Thus if the num¬ 
ber of things be 5 ; then 2 5 — 5 — 1 = 32 — & — 26. 
2. To find the number of changes and alterations which 
any number of quantities can undergo, when combined 
in all poffible varieties of waiys, with themfelves and each 
other, both as to the things themfelves, and the order or 
pofition of them. 
One thing admits but of one order or pofition. Two 
things may be varied four ways; thus, aa, ab, ba, bb. 
Three quantities, taken by twos, may be varied nine 
ways ; thus, aa, ab, ac, ba, ca, bb, be, cb, cc. In like 
manner four things, taken by twos, may be varied 4 2 or 
16 ways; and 5 things, by twos, 5 2 or 25 ways; and, in 
general, n things, taken by twos, may be changed or 
varied n- different ways. For the fame reafon, when 
taken by threes, the changes will be n *; and when taken 
by fours, they will be « 4 ; and fo generally, when taken 
by n's, the changes will be n n . Hence, then, adding all 
thefe together, the whole number of changes, or combi¬ 
nations in n things, taken both by 2’s, by 3’s, by 4’s, 
&c. to n's, will be the fum of the geometrical feries 
fl * 1 mmmm J 
k 4- n 2 + n z -f 7i 4 - - - - n n , which fum is — -- X n. 
n — 1 
For example, if the number of things n be 45 this 
4 4 — 1 255 
gives - X4 - — X 4 = 34-0- 
4 *“■ 13 
And if n be 24, the number of letters in the alphabet 1 
the theorem gives 
24 s4 — 1 - 24 
-- X 24 — 24 24 —1 X — = 
24 — 1 23 
1391724288887252999425128493402200. In fo many dif¬ 
ferent ways, therefore, may the 24 letters of the alphabet 
be varied or combined among themfelves, or fo many dif¬ 
ferent words may be made out of them. 
To COMBI'NE, v. a. [ combiner, Fr. binos j lingere, Lit.} 
To join together: 
Let us then not fufpeft our happy (late. 
As not fecure to fingle or combin’d. Milton, 
To link in union : 
God, the beft maker of all marriages, 
Combine your hearts in one, your realms in one. Shake/, 
To agree ; to accord ; to fettle by compaft: 
My heart’s dear love is fet on his fair daughter; 
As mine on hers, fo hers is fet on mine. 
And all combin'd, fave what thou mult combine 
By holy marriage. Sbakefpeare. 
To join words or ideas together: oppofed to analyfe. 
To COMBI'NE, <v. n. To coalefce; to unite each with 
other. Ufed both of things and perfons: 
Honour and policy, like unfever’d friends 
1’ th’ war, do grow together : grant that, and tell me 
In peace what each of them by th’ other lofes, 
That they combine not there ? Sbakefpeare. 
To unite in friendfhip or defign .—Combine together ’gainfl 
the enemy. Sbakefpeare. 
You with your foes combine, 
And feem your own deftruftion to defign. Dry den. 
COMBI'NE ABLE, adj. Confiftent; that may be com¬ 
bined.—I infill upon it, that pleafures are very combine- 
able with both bufinefs and ftudies, and have a much bet¬ 
ter relifh from the mixture. Cheferfield. 
COM'BLESS, adj. Wanting a comb or creft : 
What, is your creft a coxcomb ?- 
-r-A comblefs cock, fo Kate will be my hen. Sbakefpeare. 
10 F - COM'BOURGj 
