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Калькулятор перестановок/сочетаний

Рассчитайте перестановки (P) и сочетания (C) с пошаговыми решениями

Ввод значений
Вычислите количество способов выбрать r элементов из n элементов

P(n, r) = ?

Выбрать r из n элементов с учетом порядка

Перестановки и сочетания

Перестановка

P(n, r) = n! / (n - r)!

Когда порядок важен

например, ABC, ACB, BAC - все разные

Сочетание

C(n, r) = n! / (r! × (n - r)!)

Когда порядок не важен

например, ABC, ACB, BAC - все одинаковые

Примеры из реальной жизни

Примеры перестановок

  • • Создание паролей
  • • Определение 1-го, 2-го, 3-го места в гонке
  • • Рассадка мест

Примеры сочетаний

  • • Формирование команды
  • • Выбор блюд в меню
  • • Формирование комитета
History and Modern Applications of Combinatorics

Historical Development of Combinatorics

Combinatorics began in ancient India and China and has evolved into a core field of modern mathematics. From Pascal's triangle and the binomial theorem to modern graph theory and cryptography, combinatorics has played a crucial role in the development of mathematics and computer science.

Ancient and Medieval

  • • India (2nd century BC): Pingala's binomial coefficients
  • • China (11th century): Yang Hui's triangle
  • • Islamic world (12th century): Al-Karaji's combinatorics
  • • Europe (13th century): Fibonacci's combinatorial problems

Modern Era

  • • Pascal (17th century): Connecting probability and combinatorics
  • • Euler (18th century): Generating function theory
  • • 20th century: Graph theory, design theory
  • • Modern: Integration with computer science

Connection to Probability Theory

Basic Probability Calculations

Classical probability: P(A) = favorable outcomes / total outcomes

Permutation probability: probability of order-dependent events

Combination probability: probability of order-independent events

Conditional probability: probability under specific conditions

Practical Application Examples

Card games: calculating poker hand probabilities

Lottery: calculating winning probabilities

Quality control: sample inspection probabilities

Genetics: gene combination probabilities

🎯 Combinatorics Mastery Strategy

• Problem classification: First determine importance of order to distinguish permutation/combination.

• Step-by-step approach: Break complex problems into smaller units.

• Use symmetry: Find problem symmetries to simplify calculations.

• Practical application: Apply combinatorics to daily problems to build intuition.

• Programming practice: Implement combinatorial algorithms through coding.

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