Combination

In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases it is possible to count the number of combinations. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient

which can be written using factorials as whenever , and which is zero when . The set of all k-combinations of a set S is sometimes denoted by .

Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection,k-multiset, or k-combination with repetition are often used. If, in the above example, it was possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.

Combination (chess)

In chess, a combination is a sequence of moves, often initiated by a sacrifice, which leaves the opponent few options and results in tangible gain. At most points in a chess game, each player has several reasonable options from which to choose, which makes it difficult to plan ahead except in strategic terms. Combinations, in contrast to the norm, are sufficiently forcing that one can calculate exactly how advantage will be achieved against any defense. Indeed, it is usually necessary to see several moves ahead in exact detail before launching a combination, or else the initial sacrifice would not be undertaken.

Definition

In 1952/53, the editors of Shakhmaty v SSSR decided on this definition: A combination is a forced sequence of moves which uses tactical means and exploits specific peculiarities of the position to achieve a certain goal. (Golombek 1977)

Irving Chernev said:

Example

A combination is usually built out of more fundamental chess tactics such as forks, pins, skewers, undermining, discovered attacks, etc. Thus a combination must be at least three moves long, but the longer it takes to recoup the initial sacrifice, the more impressive the combination. The position below from a game between G. Stepanov and Peter Romanovsky begins a combination which illustrates several forks and skewers.

Horse jumping obstacles

Various obstacles are found in competitive sports involving horse jumping. These include show jumping, hunter, and the cross-country phase of the equestrian discipline of eventing. The size and type of obstacles vary depending on the course and the level of the horse and rider, but all horses must successfully negotiate these obstacles in order to complete a competition. Fences used in hunter and eventing are generally made to look relatively rustic and natural.

In jumping competition, they are often brightly colored and creatively designed. In hunter and jumper competition, obstacles are constructed to fall down if struck by the horse. In eventing, they are built to be solid, though for safety, certain elements may be designed to break away if hit.

Arrowhead

Also called chevrons, these fences are shaped like triangles, with the point facing towards the ground. They are generally very narrow, usually only a few feet wide. Arrowhead fences require the rider to keep their horse straight between their hands and legs, as it is easy for a run-out to occur due to the narrowness of the fence. These fences are often used in combination with other obstacles to increase their difficulty, such as right after a bank or as the second obstacle in a bending line. This tests the rider's ability to regain control of his/her horse following an obstacle.