COMBINATION. 
which appears surprizing enough, when one 
considers that two letters or figures can 
only be combined twice. See Changes. 
Combination, doctrine of. Prob. 1. Any 
number of quantities being given, together 
with the number in each combination, to 
find the number of combinations. One 
quantity admits of no combination ; two, 
a and b, only of one combination; of three 
quantities, a be, there are three combina- 
tions, viz. ab, ae, be; of four quantities, 
tliere are six combinations, viz. ab, ac, ad, 
be, bd, cd; of five quantities, there are ten 
combinations, riz. ab, ac, be, ad, bd, cd, 
ae, be, ce, de. Hence it appears that the 
numbers of combinations proceed as 1, 3, 
6 , 10 ; that is, they are triangular numbers 
whose sides differ by unity from the number 
of given quantities. If tliis then be supposed 
q, the side of the number of combinations 
will he q — 1 , and so the number of combi- 
nations See Triangular 
1 ’ 2 
NUMBERS. 
If three quantities are to be combined, 
and the number in each combination be 
three, there will be only one combination, 
abc-, if a fourth be added, four combina- 
tions will be found, abc,, a bd, bed, acd; 
if a fifth be added, the combinations will be 
ten, viz. abc, ab d, bed, acd, abe, b de, 
bee, ace, ade; if a sixth, the combina- 
tions will be twenty, &c. The number-s, 
therefore, of combinations proceed as 1, 4, 
10 , 20 , &c. that is, they are the first pyra- 
midal triangular numbers, whose side differs 
by two units from the number of given 
quantities. Hence if the number of given 
quantities be q, the side will be 5 — 2 , and 
so the number of combinations — j — ’ 
q—1 g +0 
2 ’ 3 ■ 
If four quantities are to be combined, we 
shall find the numbers of combinations to 
proceed as pyramidal triangular numbers 
of the second order, 1, 5, 15, 35, &c. whose 
side differs from tire number of quantities 
by the exponent minus an unit. Where- 
fore if the number of quantities be q, the 
side will be q — 3, and the number of com- 
q — 3 q^ q — 1 q +0 
2 ’ 3 ’ 4 ■ 
binations 
See 
PYRAMIDAL NUMBERS. 
Hence is easily deduced a general rule 
of determining the number of combinations 
in any case whatsoever. Suppose, for ex- 
ample, the number of quantities to be com- 
VOL. II. 
bined q, and the exponent of combination 
n; the number of combinations will be 
q — n-\-l q — n-|-2 q — m-}-3 q — «-f-4 
~~1 ’ 2 ’ 3 ’ 4 ’ 
&c. till the number to be added be equal 
to n. Take q = 6 and « = 4, the number of 
6 — 4 — j-" 1 6 — 4 — 1“ 2 
1 ' i 
6—3 6—2 
combinations will be 
6 — 4 + 3 6 — 4 + 4. 
2 
iii=i5. 
12 3 4 
3 4 1 
6 — 1 6 + 0 __3 
3 4 ^ 
If it be required to know all the possible 
combinations of the given quantities, be- 
ginning with the combinations of the several 
two’s, then proceeding to three’s, &c. we 
must add ^ 
q — 3 q — 2 9 — 1 9 + 0 
2 3 4 
-, &c. 
Whence the number of all the possible 
combinations will be - ^ ^ — 
12 ‘12 
9-2 ■ 9 9-1 q — 'i 
3~12 3 4 ‘12 
? ^ ^ ~ — — which is the sum of tlie 
3 4 5 
unci® of the binomial raised to the power 
q, and abridged of the exponent of the 
power increased by unity 9 + 1. Where- 
fore since these unci® come out 1 + 1 by 
being raised to the power q; and since 
1 + 1 is equal to 2 , ^i — q — 1 will be the 
number of all the possible combinations. 
For example, if the number of quantities be 
5 , the number of possible combinations will 
be 2='— 6 = 32 — 6 = 26. 
Prob. 2. Any number of quantities being 
given to find the number of all the changes 
which these quantities, combined in all the 
manners possible, can undergo. Let there 
be two quantities a and b, their variations 
wilt be two ; consequently, as each of them 
may be combined with itself, to these there 
must be added two variations more. There- 
fore the number of the whole will be 2 + 2 
= 4. If there were tliree quantities, and 
the exponent of the variation 2 , the com- 
binations will be 3, and the changes 3 ; to 
wit, ab, ac, be, and ba,ca,cb; to which if 
we add the three combinations of each 
quantity with itself a a, bb, cc, we shall 
have the number of changes 3 + 3 + 3 
= 9. 
In like manner, it is evident, if the given 
quantities were 4, and the exponent 2, that 
the numberjof combinations will be 6, and 
the number of changes likewise 6 , and the 
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