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• Combination
  ...objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size <math>r</math> from an original set of size The common forms of denoting the number of combinations of <math>{r}</math> objects from a set of <math>{n}<
  4 KB (615 words) - 10:43, 21 May 2021
• Induction
  ...at the result holds for <math>n=k+1</math>. In this way, we can show that the result holds for all positive integers; we will have shown that it works fo ...t, so does <math>n=k+2</math>. If you wish, you can similarly induct over the powers of 2.
  5 KB (768 words) - 23:59, 28 September 2024
• Binomial Theorem
  The '''Binomial Theorem''' states that for [[real]] or [[complex]] <math>a</mat ...cted are the same as the entries in the <math>n</math>th row of [[Pascal's Triangle]].
  5 KB (935 words) - 12:11, 20 February 2024
• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
  12 KB (1,993 words) - 22:47, 27 October 2024
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its various properties play a large role in [[combi ...se k}}=2^n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
  5 KB (838 words) - 16:20, 3 January 2023
• 1992 AIME Problems
  Find the sum of all positive rational numbers that are less than 10 and that have de ...>.503</math>. What's the largest number of matches she could've won before the weekend began?
  8 KB (1,117 words) - 04:32, 11 November 2023
• 1992 AIME Problems/Problem 4
  ...ch entry is the sum of the two entries above it. The first few rows of the triangle are shown below. ...ow of [[Pascal's Triangle]] do three consecutive entries occur that are in the ratio <math>3 :4 :5</math>?
  3 KB (476 words) - 13:13, 20 April 2024
• 2003 AIME I Problems/Problem 13
  ...math> representation has more <math>1</math>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</math>. ...+1</math> digits in base <math>2</math> notation, with <math>k+1</math> of the digits being <math>1</math>'s.
  4 KB (651 words) - 18:42, 7 October 2023
• 2002 AIME II Problems/Problem 12
  ...be the ratio of shots made to shots attempted after <math>n</math> shots. The probability that <math>a_{10} = .4</math> and <math>a_n\le.4</math> for all ...umber of such paths using a Pascal's Triangle type method below, computing the number of paths to each point that only move right and up.
  7 KB (1,127 words) - 12:34, 19 June 2022
• Geometry/Olympiad
  * [[Orthic triangle]] **[[Pascal's Theorem]]
  2 KB (242 words) - 09:16, 18 June 2023
• Combinatorics/Introduction
  ...comes more advanced, the scope of combinatorics grows, leading students to the [[Combinatorics/Intermediate | intermediate]] level. ** [[Pascal's triangle]]
  2 KB (159 words) - 20:39, 6 October 2023
• Power's of 2 in pascal's triangle
  == Pascal's Triangle == ...Triangle is a triangular array of numbers where each number is the sum of the two numbers above it. It Looks something like this:
  2 KB (341 words) - 15:57, 16 June 2019
• 2007 AIME II Problems/Problem 10
  ...[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0, 6, or 1, 5 elements. The total probability is <math>\frac{2}{64} + \frac{2}{64} = \frac{4}{64}</math
  8 KB (1,368 words) - 18:02, 20 July 2024
• 2007 AIME II Problems/Problem 13
  ...f <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? Label each of the bottom squares as <math>x_0, x_1 \ldots x_9, x_{10}</math>.
  4 KB (678 words) - 07:06, 18 October 2024
• Pascal Triangle Related Problems
  ==The Triangle== Here are lines zero through eight of Pascal's triangle:
  4 KB (513 words) - 19:18, 3 January 2023
• Pascal's Theorem
  </asy>|right|A diagram of the theorem}} ...]] is inscribed in a [[conic section]], then the points of intersection of the pairs of its opposite sides are collinear:
  5 KB (833 words) - 14:23, 6 July 2024
• Triangular number
  ...riangular numbers''' are the numbers <math>T_n</math> which are the sum of the first <math>n</math> [[natural number]]s from <math>1</math> to <math>n</ma The <math>n^{th}</math> triangular number is the sum of all natural numbers from one to n.
  2 KB (276 words) - 09:48, 13 August 2024
• Pascal's Identity
  '''Pascal's Identity''' is a useful theorem of [[combinatorics]] dealing with [[combi ...dentity is also known as Pascal's Rule, Pascal's Formula, and occasionally Pascal's Theorem.
  3 KB (496 words) - 14:43, 11 April 2024
• 2008 Mock ARML 2 Problems
  ...>. Given that <math>m\angle{ABC} + m\angle{BCD} = 270^{\circ}</math>, find the area of <math>ABCD</math>. ...rs, each of which is less than <math>30</math>, is <math>2397</math>, find the sum of all positive integers with exactly three proper divisors, each of wh
  4 KB (582 words) - 20:57, 8 May 2019
• 1995 AHSME Problems/Problem 27
  ...sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown. ...h>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?
  5 KB (682 words) - 08:45, 18 February 2022

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