Lesson 1.12: Permutation, Combination & Probability

Introduction:
Permutation, Combination, and Probability are important topics in SSC exams. These concepts help in solving questions related to arrangements, selections, and likelihood of events.

1. Permutation (Arrangement)

• Definition: Number of ways of arranging objects in order.

• Formula:

nPr=n!(n−r)!^nP_r = \frac{n!}{(n-r)!}nPr​=(n−r)!n!​

Where: n = total objects, r = objects selected, n! = n × (n-1) × … × 1

• Example:
  Arrange 3 letters (A, B, C) → 3! = 6 ways
  (ABC, ACB, BAC, BCA, CAB, CBA)

2. Combination (Selection)

• Definition: Number of ways of selecting objects without order.

• Formula:

nCr=n!r!(n−r)!^nC_r = \frac{n!}{r!(n-r)!}nCr​=r!(n−r)!n!​

• Example:
  Select 2 letters from A, B, C → 3C2=3!2!1!=3^3C_2 = \frac{3!}{2!1!} = 33C2​=2!1!3!​=3 ways
  (AB, AC, BC)

3. Probability

• Definition: Likelihood of an event happening.

• Formula:

P(E)=Number of favorable outcomesTotal number of outcomesP(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}P(E)=Total number of outcomesNumber of favorable outcomes​

• Example:
  Toss a die. Probability of getting 4 = 1/6

• Rules:

  • 0 ≤ P(E) ≤ 1

  • Sum of probabilities of all events = 1

4. SSC Exam Tips

• Use factorials (!) to simplify calculations.

• For repeated elements, divide by factorial of repetitions:

n!p!q!r!\frac{n!}{p!q!r!} p!q!r!n!​

• Practice questions with dice, coins, cards, and balls.

• Memorize shortcuts for small values of n and r.

5. Practice Questions

1. Arrange 4 letters A, B, C, D. How many ways?

2. Select 3 students from 5. How many combinations?

3. Toss two coins. Probability of getting at least one head?

4. In a bag, 5 red and 3 blue balls. Probability of picking a red ball?

5. How many ways to arrange the letters of “LEVEL”?