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• Wilson's Theorem
  ...h> does not divide <math>(p-1)! + 1</math>. Therefore <math>p</math> does not divide <math>(p-1)! + 1</math>. ...th>a(b-c)</math> is a multiple of <math>p</math>, when <math>p</math> does not divide <math>a</math> or <math>b-c</math>&mdash;a contradiction.) This inv
  4 KB (639 words) - 01:53, 2 February 2023
• Principle of Inclusion-Exclusion
  ...A_2|</math> at least once. However, some things were counted twice. The elements that were counted twice are precisely those in <math> {}A_1\cap A_2 </math> ...ted them twice originally, and just subtracted them out once. However, the elements that are in all three sets were originally counted three times and then sub
  9 KB (1,703 words) - 07:25, 24 March 2024
• Combination
  ...ame as <math>ACB</math>) but in combinations the order of arrangement does not matter(<math>ABC</math> is equivalent to <math>ACB</math>).For its derivati ...se <math>r</math> elements are selected out.For permutation <math>r</math> elements can be arranged in <math>r!</math> ways.We have overcounted the number of c
  4 KB (615 words) - 11:43, 21 May 2021
• Fermat's Little Theorem
  ...[integer]], <math>{p}</math> is a [[prime number]] and <math>{a}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 ...cause we no longer need to restrict ourselves to integers <math>{a}</math> not divisible by <math>{p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Induction
  ...binatorics]] are especially susceptible to induction solutions, but that's not to say that there aren't any problems in other areas, such as [[Inequalitie ...onempty subsets <math>S</math> of <math>\{1, 2, . . . , n\}</math> that do not contain two consecutive integers. Prove that the sum of all the f(S)’s of
  5 KB (768 words) - 20:45, 1 September 2022
• Overcounting
  ...ing more than desired and then systematically "correcting" for overcounted elements by removing them from the total count via subtraction or division. The idea (Note that it is not a coincidence that 67 is close to two-thirds of 100! We can approach this p
  4 KB (635 words) - 12:19, 2 January 2022
• Cardinality
  ...or showed that there are multiple infinite cardinalities. In other words, not all infinite sets are the same size.
  2 KB (263 words) - 00:54, 17 November 2019
• Geometry
  ...lly outlined by the Greek [[mathematician]] [[Euclid]] in his book ''[[The Elements]]''. :''“Through any line and a point not on the line, there is exactly one line passing through that point parallel
  3 KB (393 words) - 07:59, 25 September 2020
• Algebraic topology
  ...opy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>. ...up of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</
  3 KB (479 words) - 15:35, 1 December 2015
• Uncountable
  ...to\mathbb{N}</math>. Assuming the [[Axiom of choice]], every set that is ''not'' uncountable is either [[finite]] or [[countably infinite]]. The most com ...ath>\{\omega_1, \omega_2, \omega_3, ...\}</math> be any enumeration of the elements of <math>A</math> (in other words, take an injection <math>f: A \to \mathbb
  2 KB (403 words) - 20:53, 13 October 2019
• Modular arithmetic/Intermediate
  ...{n}</math> allows us to divide the set of integers into sets of equivalent elements. For example, if <math>n = 3</math>, then the integers are divided into th ...lled a [[ring]] -- a structure in which we can add, subtract, and multiply elements.
  14 KB (2,317 words) - 19:01, 29 October 2021
• Limit
  ...s the value to which a function grows close when its argument is near (but not at!) a particular value. For example, ...but not at!) the value in question: when <math>x</math> is very close (but not equal!) to zero, <math>f(x)</math> will always be close to (in fact equal t
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems
  ...> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> ...ath> where <math> a, b, c </math> are distinct digits. Find the sum of the elements of <math> \mathcal{S}. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 13
  ...ath> elements of <math>S</math> that are divisible by <math>2^1</math> but not by <math>2^2</math>. ...ath> dividing <math>S_n</math> has an odd exponent, so <math>S_n</math> is not a perfect square.
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2004 AMC 12B Problems
  ...>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...ath>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits? Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12A Problems/Problem 25
  How many ways are there to choose <math>k</math> elements from an ordered <math>n</math> element [[set]] without choosing two consecu ...This converts your configuration into a configuration with <math>k</math> elements where the largest possible element is <math>n-k+1</math>, with no restricti
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12B Problems/Problem 20
  Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set <math>\{-7,-5,-3,-2,2,4,6,13\}.</math> ...ther have to be 4 odds, 4 evens, or 2 odds and evens. The 4 odd numbers do not work (38) nor the 4 even numbers (26).
  3 KB (463 words) - 19:28, 6 November 2022
• 2005 AMC 12B Problems/Problem 25
  ...tex. In this case, the ant has visited three corners of a square, but can not next visit the fourth corner or there will be no way to connect the remaini ...math>. So the problem is analogous to finding the number of ways to take 2 elements from each of these sets to make the set <math>\{A,B,C,D,E,F\}</math>.
  10 KB (1,840 words) - 21:35, 7 September 2023
• Set
  ...et," like ''collection, ensemble, group'', etc., but those names really do not define the meaning of the word ''set''; all they can do is replace it in va ...the set. A common misconception is that a set can have multiple indistinct elements, such as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entit
  11 KB (2,021 words) - 00:00, 17 July 2011

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