Distribution Problems — Identical/distinct objects into boxes
Permutation & Combination
9
JEE Qs
8%
Hard
75
min
Identify the nature of objects (identical/distinct) and boxes (identical/distinct) first, then apply the appropriate method or formula, carefully considering all constraints.
🧮 Key Formulas
✅ Key Points for JEE
- 1Carefully distinguish between identical and distinct objects, and identical and distinct boxes, as each combination requires a different approach.
- 2For distributing identical objects into distinct boxes, the 'Stars and Bars' method is fundamental. Remember its two forms: (n+r-1) C (r-1) for empty boxes allowed, and (n-1) C (r-1) for no empty boxes.
- 3When distributing distinct objects into distinct boxes, each object has 'r' independent choices, leading to r^n total ways.
- 4Problems involving identical boxes (for either distinct or identical objects) are generally more complex, often requiring the use of Stirling Numbers of the Second Kind for distinct objects or Partition of an Integer for identical objects. These are frequently solved by enumeration or case analysis for small values in JEE.
- 5Constraints like 'at least one object in each box' or 'maximum capacity' often require a combination of direct formulas with inclusion-exclusion principles or specific case work.
⚠️ Common Mistakes
- ✕Confusing identical objects/boxes with distinct objects/boxes, leading to incorrect formula application (e.g., using stars and bars for distinct objects).
- ✕Not correctly applying the 'no empty boxes' constraint, especially with stars and bars, or forgetting that (n-1) C (r-1) requires n >= r.
- ✕Errors in calculating combinations or permutations, particularly with factorials and binomial coefficients.
📝 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 Maths Ch 7: Permutations and Combinations