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The number of such strings is the number of ways to place 10 stars in 13 positions, () = =, which is the number of 10-multisubsets of a set with 4 elements. Bijection between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right).
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.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
5; 4 + 1; 3 + 2. Note that the ancient Sanskrit sages discovered many years before Fibonacci that the number of compositions of any natural number n as the sum of 1's and 2's is the nth Fibonacci number! Note that these are not general compositions as defined above because the numbers are restricted to 1's and 2's only. 1=1 (1) 2=1+1=2 (2)
For instance, the number 25 in column k = 3 and row n = 5 is given by 25 = 7 + (3×6), where 7 is the number above and to the left of 25, 6 is the number above 25 and 3 is the column containing the 6.
Rather, as explained under combinations, the number of n-multicombinations from a set with x elements can be seen to be the same as the number of n-combinations from a set with x + n − 1 elements. This reduces the problem to another one in the twelvefold way, and gives as result
No. 32 appeared most often — 173 times — among the first five balls drawn in winning combinations, followed by the No. 39 in 163 combinations, according to data.
This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 85 5 is only slightly bigger than 2 32. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.