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One must divide the number of combinations producing the given result by the total number of possible combinations ... 0.0000180208 55,491.33 5 + 1
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, 0-1-2 and 0-2-3 are possible, but 0-2-1 is not allowed. Given these restrictions, with the six potential values (0–5) commonly seen, there are 56 unique combinations, and thus the standard triomino set has 56 tiles. Larger sets are possible; for example, including 6 as a possible end number would result in 84 tiles.
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 , including the empty set:
For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements. For example, if n = 10 and k = 4, the theorem gives the number of solutions to x 1 + x 2 + x 3 + x 4 = 10 (with x 1, x 2, x 3, x 4 > 0) as the binomial coefficient
For each of the n − 1 hats that P 1 may receive, the number of ways that P 2, ..., P n may all receive hats is the sum of the counts for the two cases. This gives us the solution to the hat-check problem: Stated algebraically, the number ! n of derangements of an n -element set is ! n = ( n − 1 ) ( !
Another example is attempting to make 40 US cents without nickels (denomination 25, 10, 1) with similar result — the greedy chooses seven coins (25, 10, and 5 × 1), but the optimal is four (4 × 10). A coin system is called "canonical" if the greedy algorithm always solves its change-making problem optimally.
Frobenius coin problem with 2-pence and 5-pence coins visualised as graphs: Sloping lines denote graphs of 2x+5y=n where n is the total in pence, and x and y are the non-negative number of 2p and 5p coins, respectively.