combination

Compare this with the permutations of the same size from the same set. Note that each combination appears twice in the list of permutations. This is because there are 2 ways to arrange two numbers. The number of combinations equals the number of permutations of the type desired divided by the number of permutations of a set that size, using all the elements.

The number of combinations of r objects from a set of n objects is often written as C(n,r) or _nC_r or with two numbers between tall parentheses sorta like this:

  /   \
 |  n  |
 |     |
 |  r  |
  \   /

C(n,r) can be calculated as n!/{r!(n-r)!} where ! represents the factorial function. Note that C(n,r) = C(n,n-r); there is a one-to-one correspondence between subsets of size r and subsets of size n-r -- just consider what's left over after you take away a subset of size r.

The combination falls within the realm of tactics, just as position play falls within the realm of strategy. But when a master plays there is an intimate relationship between the two, neither of which exists independant of the other. Proper understanding of combination play, and an ability to calculate combinations, is neccessary to an understanding of the nature of position play. Conversely, no combination is valid that does not arise from positional considerations. A queen sacrifice leading to mate can not be achieved unless the sacrifice is justified by the position, and the player who has a better understanding of position play is most likely to achieve the sort of advantage that can lead to such winning combinations.

Emanuel Lasker in his famous Manual of Chess, described how a combination arises from a the position, as follows:

"In his analysis of the position the Chess-player
has to see through the maze of variants in order
to ascertain whether on not by forceful moves the
game may be brought at once to a conclusion.  He 
makes these investigations often, he makes them
always when the hostile forces have approached
each other for mortal combat.  Otherwise, he might
let slip a favourable opportunity or fail in
vigilance and lose thereby in a moment of inattention
what he has gained in an hour of concentration."

Not every imagined sequence of moves that goes into the calculation of a combination will materialize on the board. In fact, most variations, and potential combinations, occur only within the mind of the player. Again, from Lasker's Manual of Chess:

"Nevertheless, ordinarily his work is merely
prophylactic.  Ordinarily, his labor remains a mere
attempt, his investigation is ordinarily not made
visible by his deed.  The move that he chooses
reflects none of his preparatory work because it
is seldom that the net of variants contains a
satisfactory solution."

"In the rare instances that the player can detect
a variation or net of them which leads to a
desirable issue by force, the totality of these
variations and their logical connections, their
structure, are named a 'combination.' And he who
follows in his play such a chain of moves is said
to 'make a combination.'"

Com`bi*na"tion (?), n. [LL. combinatio. See Combine.]

1.

The act or process of combining or uniting persons and things.

Making new compounds by new combinations. Boyle.

A solemn combination shall be made Of our dear souls. Shak.

2.

The result of combining or uniting; union of persons or things; esp. a union or alliance of persons or states to effect some purpose; -- usually in a bad sense.

A combination of the most powerful men in Rome who had conspired my ruin. Melmoth.

3. Chem.

The act or process of uniting by chemical affinity, by which substances unite with each other in definite proportions by weight to form distinct compounds.

4. pl. Math.

The different arrangements of a number of objects, as letters, into groups.

⇒ In combinations no regard is paid to the order in which the objects are arranged in each group, while in variations and permutations this order is respected.

Brande & C.

Combination car, a railroad car containing two or more compartments used for different purposes. [U. S.] -- Combination lock, a lock in which the mechanism is controlled by means of a movable dial (sometimes by several dials or rings) inscribed with letters or other characters. The bolt of the lock can not be operated until after the dial has been so turned as to combine the characters in a certain order or succession. -- Combination room, in the University of Cambridge, Eng., a room into which the fellows withdraw after dinner, for wine, dessert, and conversation. -- Combination by volume Chem., the act, process, or ratio by which gaseous elements and compounds unite in definite proportions by volume to form distinct compounds. -- Combination by weight Chem., the act, process, or ratio, in which substances unite in proportions by weight, relatively fixed and exact, to form distinct compounds. See Law of definite proportions, under Definite.

Syn. -- Cabal; alliance; association; league; union; confederacy; coalition; conspiracy. See Cabal.

© Webster 1913.