Fundamental Principle — Counting techniques
Permutation & Combination
9
JEE Qs
8%
Hard
60
min
Always clearly define the 'stages' or 'cases' in a problem and decide whether they are sequential (AND - multiply) or alternative (OR - add) to correctly apply the fundamental principles.
🧮 Key Formulas
✅ Key Points for JEE
- 1The 'OR' keyword or alternative choices in problems generally implies the use of the Addition Principle for mutually exclusive events.
- 2The 'AND' keyword or sequential steps involved in completing a task imply the use of the Multiplication Principle.
- 3Always address conditions or restrictions (e.g., 'no repetition allowed', 'must start with a vowel') first, as they often limit the choices available in certain steps.
- 4Break down complex problems into smaller, manageable stages or mutually exclusive cases, applying the appropriate principle to each, and then combining results.
- 5Be careful to avoid overcounting (if cases overlap and are not strictly mutually exclusive) or undercounting by missing possible cases.
⚠️ Common Mistakes
- ✕Incorrectly applying the Addition Principle when events are sequential or dependent, or the Multiplication Principle when events are mutually exclusive.
- ✕Failing to identify and properly handle all given constraints or conditions, leading to an incorrect number of choices in various steps.
- ✕Not considering all possible mutually exclusive cases when solving a problem, especially when the wording implies 'at least one' or 'either/or' scenarios.
📝 Practice Questions
See allQ2. In a group of 3 girls and 4 boys, there are two boys B1 and B2 . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but B1 and B2 are not adjacent to each other, is : (1) 96 (2) 144 (3) 120 (4) 72
Q18.Let the shortest distance from (a, 0), a > 0, to the parabola y2 = 4x be 4 . Then the equation of the circle passing through the point (a, 0) and the focus of the parabola, and having its centre on the axis of the parabola is : (1) x2 + y2 −10x + 9 = 0 (2) x2 + y2 −6x + 5 = 0 (3) x2 + y2 −4x + 3 = 0 (4) x2 + y2 −8x + 7 = 0
Q25.The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is______.
Q10.From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is : 2025 (22 Jan Shift 1) JEE Main Previous Year Paper (1) 5148 (2) 6084 (3) 4356 (4) 14950
Q13.The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is (1) 36000 (2) 37000 (3) 34000 (4) 35000
Q23.The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is -
NCERT Chapters
- Class 11 Mathematics Ch 7: Permutations and Combinations