PERMUTATIONS AND COMBINATIONS

Permutations and Combinations Study Summary

This document provides a comprehensive overview of Permutations and Combinations, including fundamental principles, definitions, formulas, important identities, and a set of practice questions with answers.

Key Concepts

Fundamental Principles of Counting

• Multiplication Principle: If an event can occur in 'm' different ways, followed by another event that can occur in 'n' different ways, then the total number of different ways of occurrence of the two events in order is m × n.
• Addition Principle: If there are two events such that they can occur independently in 'm' and 'n' different ways respectively, then either of the two events can occur in (m + n) ways.

Factorial

• Definition: The factorial of a natural number 'n', denoted by n! or ⌥n, is the continued product of the first 'n' natural numbers.

  n! = n × (n - 1) × (n - 2) × ......... × 3 × 2 × 1

• Properties:
  • n! = n × ((n - 1)!)
  • n! = n × (n - 1) × ((n - 2)!)

Permutations

• Definition: A permutation is an arrangement of a number of objects in a definite order, taken some or all at a time.
• Number of Permutations of 'n' different objects taken 'r' at a time (without repetition):

  ⁿP_r = P(n, r) = n! / (n-r)!

  (where 0 ≤ r ≤ n)

• Number of Permutations of 'n' objects when repetition is allowed:

  n^r

• Number of Permutations of 'n' objects where p₁ are of one kind, p₂ are of second kind, ..., p_k are of k^th kind and the rest are different:

  n! / (p₁! p₂! ... p_k!)

Combinations

• Definition: A combination is a selection made by choosing some or all of a number of objects, without considering their order.
• Number of Combinations of 'n' objects taken 'r' at a time:

  ⁿC_r = C(n, r) = n! / (r! (n-r)!)

  (where 0 ≤ r ≤ n)

Important Identities and Results

• 0! = 1
• ⁿC₀ = ⁿC_n = 1
• ⁿC_r = ⁿC_n-r (where 0 ≤ r ≤ n, and r are positive integers)
• ⁿP_r = r! × ⁿC_r (where 0 ≤ r ≤ n, r and n are positive integers)
• ⁿC_r + ⁿC_r+1 = ⁿ⁺¹C_r+1 (where 0 ≤ r ≤ n and r and N are positive integers)
• If ⁿC_a = ⁿC_b, then either a = b or a + b = n

Practice Questions

The document includes a variety of practice questions categorized by marks and type:

• Section A: Very Short Answer Type Questions (1 Mark)
• Section B: Short Answer Type Questions (2 Marks)
• Section C: Short Answer Type Questions (4 Marks)
• Section D: Long Answer Type Questions (6 Marks)

Answers

Detailed answers are provided for all the practice questions at the end of the document.