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The numerator equates to the number of ways to select the winning numbers multiplied by the number of ways to select the losing numbers. For a score of n (for example, if 3 choices match three of the 6 balls drawn, then n = 3), ( 6 n ) {\displaystyle {6 \choose n}} describes the odds of selecting n winning numbers from the 6 winning numbers.
The number of k-combinations for all k is the number of subsets of a set of n elements. There are several ways to see that this number is 2 n . In terms of combinations, ∑ 0 ≤ k ≤ n ( n k ) = 2 n {\textstyle \sum _{0\leq {k}\leq {n}}{\binom {n}{k}}=2^{n}} , which is the sum of the n th row (counting from 0) of the binomial coefficients in ...
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 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
The Stirling numbers of the second kind can represent the total number of rhyme schemes for a poem of n lines. (,) gives the number of possible rhyming schemes for n lines using k unique rhyming syllables. As an example, for a poem of 3 lines, there is 1 rhyme scheme using just one rhyme (aaa), 3 rhyme schemes using two rhymes (aab, aba, abb ...
The following is an example of an abbreviated wheeling system for a pick-6 lottery with 10 numbers, 4 if 4 guarantee, and the minimum possible number of combinations for that guarantee (20). A template for an abbreviated wheeling system is given as 20 combinations on the numbers from 1 to 10.
The Frobenius number exists as long as the set of coin denominations is setwise coprime. There is an explicit formula for the Frobenius number when there are only two different coin denominations, and , where the greatest common divisor of these two numbers is 1: . If the number of coin denominations is three or more, no explicit formula is known.
The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. This formula can also be stated in a recursive form.