In mathematics a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, say an apple, orange and 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 can refer to the combination of n things taken k at a time without or with repetitions. In the above example repetitions were not allowed. If however it was possible to have two of any one kind of fruit there would be 3 more combinations: one with two apples, one with two oranges, and one with two pears.
With large sets, it becomes necessary to use more sophisticated mathematics to find the number of combinations. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.
Famous quotes containing the word combination:
“Just as we need to encourage women to test lifes many options, we need to acknowledge real limits of energy and resources. It would be pointless and cruel to prescribe role combination for every woman at each moment of her life. Life has its seasons. There are moments when a woman ought to invest emotionally in many different roles, and other moments when she may need to conserve her psychological energies.”
—Faye J. Crosby (20th century)
“The principle of fashion is ... the principle of the kaleidoscope. A new year can only bring us a new combination of the same elements; and about once in so often we go back and begin again.”
—Katharine Fullerton Gerould (18791944)
“[N]o combination of dictator countries of Europe and Asia will halt us in the path we see ahead for ourselves and for democracy.... The people of the United States ... reject the doctrine of appeasement.”
—Franklin D. Roosevelt (18821945)