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These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2 n − 1, where each digit position is an item from the set of n. Given 3 cards numbered 1 to 3, there are 8 distinct combinations ( subsets ), including the empty set :
The number associated in the combinatorial number system of degree k to a k-combination C is the number of k-combinations strictly less than C in the given ordering. This number can be computed from C = {c k, ..., c 2, c 1} with c k > ... > c 2 > c 1 as follows.
For example, when d=4, the hash table for two occurrences of d would contain the key-value pair 8 and 4+4, and the one for three occurrences, the key-value pair 2 and (4+4)/4 (strings shown in bold). The task is then reduced to recursively computing these hash tables for increasing n , starting from n=1 and continuing up to e.g. n=4.
As the number of bits composing a string increases, the number of possible 0 and 1 combinations increases exponentially. A single bit allows only two value-combinations, two bits combined can make four separate values, three bits for eight, and so on, increasing with the formula 2 n. The amount of possible combinations doubles with each binary ...
The list of all single-letter-single-digit combinations contains 520 elements ... A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 A-0 A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 B0 B1 B2 B3 B4 ...
As an numeric example how many combinations can 3 pairs of brackets be legally arranged? From the Binomial interpretation there are ( 2 m m ) {\displaystyle {\tbinom {2m}{m}}} or numerically ( 6 3 ) {\displaystyle {\tbinom {6}{3}}} = 20 ways of arranging 3 open and 3 closed brackets.
The number 2982 is 4:0:4:1:0:0:0! in factoradic, and that number picks out digits (4,0,6,2,1,3,5) in turn, via indexing a dwindling ordered set of digits and picking out each digit from the set at each turn:
The list of all single-letter-double-digit combinations contains 7,800 elements of the form [[{{letter}}{{digit}}{{digit}}]] and [[{{letter}}-{{digit}}{{digit}}]] and ...