Definition and Calculation of Combinations
– A combination is a selection of items from a set with distinct members.
– The order of selection does not matter.
– A combination is a subset of distinct elements of a set.
– Two combinations are identical if they have the same members.
– The arrangement of members in each set does not matter.
– The number of combinations, denoted by C(n, k), is equal to the binomial coefficient.
– The formula for C(n, k) is n! / (k!(n-k)!).
– The formula can be derived from the fact that each combination of a set has k! permutations.
– The number of combinations is zero when k > n.
– The set of all combinations of a set is often denoted by {S choose k}.

Combinations with Repetition
– Combinations in which repetition is allowed are called combinations with repetition.
– They are also referred to as multisets or selections.
– For example, if there are two of any one kind of fruit, there would be additional combinations.
– Combinations with repetition are used when counting possibilities in various scenarios.

Counting Combinations in Practical Examples
– Large sets make it impractical to write a complete list of combinations.
– For example, a poker hand can be described as a 5-combination of cards from a 52-card deck.
– There are 2,598,960 possible combinations of a 5-card hand.
– The chance of drawing any one hand at random is 1/2,598,960.
– Combinations can be computed explicitly using various methods.

Properties and Symmetry of Combinations
– Binomial coefficients exhibit properties and symmetry.
– The formula for combinations can be used to compute adjacent numbers in Pascal’s triangle.
– There are relations between adjacent numbers in Pascal’s triangle in all three directions.
– Combinations have a symmetry property: C(n, k) = C(n, n-k).
– A formula that exhibits this symmetry directly is C(n, k) = n! / (k!(n-k)!).

Enumeration of Combinations
– Adding a new largest element does not change the initial part of the enumeration.
– New combinations are added after the previous ones.
– The enumeration can be extended indefinitely with combinations of larger sets.
– The combinatorial number system is a bijection that computes the combination at a given place in the enumeration.
– It is also known as rank/ranking and unranking in computational mathematics.
– One way to enumerate combinations is by visiting all binary numbers less than 2^n.
– Choosing numbers with nonzero bits is inefficient for large n.
– Another way is to track index numbers of the selected elements.
– The first allowed combination is {0, 1, …, k-1} or {1, 2, …, k}.
– The next allowed combination is obtained by incrementing the last index number or resetting the index numbers.Sources: