Combinatorial Mathematics

Mathematician Ivan Niven subtitled his 1965 book on combinatorial mathematics, Mathematics of Choice, with the phrase, How to Count without Counting. Determining the quantity can always be done by counting - increasing the total by one for each item or instance - but when the number is huge, counting takes a long time. Counting the number of substitution schemes in a monoalphabetic cipher, for example, is a prohibitively time-consuming task (and a boring one).

Combinatorial mathematics, or combinatorics, as it is often called, is a field of mathematics concerned with the arrangement or order of a set of objects. Consider the number of schemes in the monoalphabetic cipher. In the process of selecting a scheme, the cryptographer takes the first letter, say a, and chooses its substitution. In the English alphabet there are 26 letters, so there are 26 possible substitutions (including a). After this choice is made, the cryptographer moves to the second letter. Since one letter has already been selected, there are 25 remaining letters to choose from. After this selection, the next letter is chosen from 24 possibilities, and so forth, with the nth letter having 27 - n possibilities. Even if the cryptographer adopts a scheme such as a shift or some other simple operation, the concept and the result are the same - selections from a number of possibilities.

A basic principle of combinatorics is that the total number of something equals the product of the number of possibilities. For example, if a person has a choice of two different shirts - red (r) and blue (b) - and three different pairs of pants - gray (g), brown (w), and blue (b) - there are 2 x 3 = 6 total wardrobe combinations - r-g, r-w, r-b, b-g, b-w, and b-b. (Some of these combinations would fail to be fashionable!)

For a 26-letter cipher, the cryptographer has 26 possibilities for the first choice, 25 for the second, 24 for third, and so on. The product of these numbers is 26 x 25 x 24 x 23 x 3 x 2 x 1. The product of all positive integers below n is called n factorial, represented mathematically by an exclamation mark - n!. Factorials are huge for even small n. For example, 26! is approximately 4.0 x 1026. It includes all schemes, including the undesirable one (from a cryptographer's point of view) of choosing a for a, b for b, and so on for each letter of the alphabet. More advanced methods of combinatorics deal with problems of selection, but even without these poor choices, the number of ways of picking a monoalphabetic cipher is enormous.

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