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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 ...
(this associates distinct numbers to all finite sets of natural numbers); then comparison of k-combinations can be done by comparing the associated binary numbers. In the example C and C′ correspond to numbers 1001011001 2 = 601 10 and 1010001011 2 = 651 10, which again shows that C comes before C′.
The number of representations of a natural number n as the sum of four squares of integers is denoted by r 4 (n). Jacobi's four-square theorem states that this is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function ), i.e.
The 1,000th Wordle puzzle ran in March, a milestone for the popular game where players tend to stick to a formula, with 2.8 million people using the same starting word every day as proof.
A staffer for a U.S. House of Representatives Democrat was arrested at the Capitol on Monday when a regular security check turned up ammunition in his bag, the Capitol Police said. The police ...
Markings typically appear every 10 smoots (56 ft; 17 m), but additional marks appear at other numbers in between. For example, the 70-smoot (390 ft; 120 m) mark is accompanied by a mark for 69 . The 182.2-smoot (1,017 ft 3 in; 310.1 m) mark is accompanied by the words "Halfway to Hell" and an arrow pointing towards MIT.
The College Football Playoff is on a break, leaving bowl season to the programs that did not make the field .A look at the games on Dec. 23.
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.