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• Principle of Inclusion-Exclusion
  ...[[union]] of a given group of [[set]]s, the size of each set, and the size of all possible [[intersection]]s among the sets. Here, we will illustrate how PIE is applied with various numbers of sets.
  9 KB (1,703 words) - 00:20, 7 December 2024

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• Combinatorics
  ...on-[[recursive]] formula for the [[Fibonacci numbers]], and so too methods of solving the [[Rubik's cube]]. Mathematicians who spend their careers studyi ...ly in [[theoretical computer science]], [[statistics]], and various fields of science.
  1 KB (209 words) - 17:13, 27 December 2024
• Principle of Inclusion-Exclusion
  ...[[union]] of a given group of [[set]]s, the size of each set, and the size of all possible [[intersection]]s among the sets. Here, we will illustrate how PIE is applied with various numbers of sets.
  9 KB (1,703 words) - 00:20, 7 December 2024
• Overcounting
  ...y important counting tools such as [[combinations]] and the [[Principle of Inclusion-Exclusion]]. An example of a classic problem is as follows:
  4 KB (635 words) - 11:19, 2 January 2022
• Combinatorics/Intermediate
  * [[Principle of Inclusion-Exclusion]] * [[Pigeonhole Principle]]
  910 bytes (77 words) - 15:23, 18 May 2021
• 2005 AIME II Problems/Problem 4
  Find the number of [[positive integer]]s that are divisors of at least one of <math> 10^{10},15^7,18^{11}. </math> ...two or more of our three numbers. Thus, we must subtract off the divisors of their pair-wise [[greatest common divisor]]s.
  3 KB (377 words) - 17:36, 1 January 2024
• 2004 AIME I Problems/Problem 8
  Define a regular <math> n </math>-pointed star to be the union of <math> n </math> line segments <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> * the points <math> P_1, P_2,\ldots, P_n </math> are coplanar and no three of them are collinear,
  4 KB (620 words) - 20:26, 5 June 2021
• 1983 AIME Problems/Problem 7
  ...f <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? We can use [[complementary counting]], by finding the probability that none of the three knights are sitting next to each other and subtracting it from <m
  9 KB (1,458 words) - 21:34, 2 February 2025
• 1984 AIME Problems/Problem 11
  ...he oak and maple trees and you also multiply the denominator by the number of ways to arrange the oak and maple trees, making them cancel out.) ...mongst the seven previous trees. We can think of these trees as 5 dividers of 8 slots that the birch trees can go in, making <math>{8\choose5} = 56</math
  7 KB (1,115 words) - 23:52, 6 September 2023
• 1990 AIME Problems/Problem 1
  ...nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...math>n</math> is an integer greater than 500 and <math>T</math> is the set of numbers which contains all <math>k^2,k^3\le 500</math>.
  2 KB (283 words) - 22:11, 25 June 2023
• 2000 AIME I Problems/Problem 5
  ...er of marbles in the two boxes is <math>25.</math> One marble is taken out of each box randomly. The [[probability]] that both marbles are black is <math ...Inclusion-Exclusion]] still requires us to find the individual probability of each box.
  7 KB (1,011 words) - 19:09, 4 January 2024
• 2002 AIME I Problems/Problem 1
  Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three ...c 1{10}</math>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindrome.
  4 KB (538 words) - 19:10, 16 September 2024
• 2001 AIME II Problems/Problem 9
  ...ach square, either color is equally likely to be used. The [[probability]] of obtaining a grid that does not have a 2-by-2 red square is <math>\frac {m}{ We can use [[complementary counting]], counting all of the colorings that have at least one red <math>2\times 2</math> square.
  8 KB (1,214 words) - 22:21, 9 January 2025
• 2001 AIME II Problems/Problem 2
  ...o could study both languages, and let <math>M</math> be the largest number of students who could study both languages. Find <math>M-m</math>. ...t\lfloor 40\% \cdot 2001 \right\rfloor = 800</math>. By the [[Principle of Inclusion-Exclusion]],
  2 KB (252 words) - 23:54, 9 January 2024
• Combinatorics/Olympiad
  ...inatorics/Intermediate | intermediate combinatorics]] along with knowledge of the following topics. * [[Pigeonhole principle]]
  705 bytes (64 words) - 15:22, 18 May 2021
• 2007 AIME I Problems/Problem 10
  ...and three shaded squares in each column. Let <math>N</math> be the number of shadings with this property. Find the remainder when <math>N</math> is div ...ut loss of generality let them be the first three rows. (Change the order of the rows to make this true.) We will multiply whatever answer we get by 20
  14 KB (2,337 words) - 08:37, 10 January 2025
• 2005 AMC 12A Problems/Problem 18
  ...3,5}\},</math> <math> \{\mathrm{prime-looking}\}</math>. Hence, the number of prime-looking numbers is <math>1000 - (168-3) - 1 - |S_2 \cup S_3 \cup S_5| ...cup S_5</math> using the [[Principle of Inclusion-Exclusion]]: (the values of <math>|S_2| \ldots</math> and their intersections can be found quite easily
  2 KB (277 words) - 17:15, 25 November 2020
• 2005 AMC 12A Problems/Problem 19
  ...proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads <tt>002005</tt>, how many miles has the ...<math>20</math> for both the other two [[intersection]]s. The intersection of all three sets is just <math>2</math>. So we get:
  4 KB (536 words) - 17:50, 26 November 2024
• 2006 AIME II Problems/Problem 7
  Find the number of [[ordered pair]]s of [[positive]] [[integer]]s <math> (a,b) </math> such that <math> a+b=1000 </ ...s into the above category, so we do not have to worry about [[Principle of Inclusion-Exclusion|overcounting]]).
  7 KB (1,114 words) - 02:41, 12 September 2021
• 2002 AMC 12B Problems/Problem 16
  ...hat is the [[probability]] that the product of the two rolls is a multiple of 3? ...P(b) = \frac{2}{6} = \frac 13</math> that Amal does. By the [[Principle of Inclusion-Exclusion]],
  2 KB (317 words) - 09:26, 5 November 2023
• 1998 AHSME Problems/Problem 24
  ...be any of the ten decimal digits <math>0,1,2, \ldots, 9</math>, the number of different memorable telephone numbers is ...line{d_1d_2d_3}=\overline{d_5d_6d_7}</math> with the exception of the case of <math>d_4=d_5=d_6=d_7</math>, which only gives one sequence. After accounti
  2 KB (330 words) - 09:14, 10 August 2016

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