Combinations — nCr, properties
Permutation & Combination
9
JEE Qs
8%
Hard
75
min
Master the art of distinguishing between selection and arrangement problems; this is the core skill for P&C.
🧮 Key Formulas
✅ Key Points for JEE
- 1Combinations deal with selection of items where the order of selection does not matter; if order matters, it's a permutation.
- 2The symmetry property nCr = nC(n-r) is highly useful for simplifying calculations and solving equations involving combinations.
- 3Pascal's Identity, nCr + nC(r-1) = (n+1)Cr, is crucial for summing consecutive combinations or for identity proofs.
- 4The total number of ways to select any number of items from 'n' distinct items (including selecting none) is 2^n.
- 5Carefully identify 'n' (total items available) and 'r' (items to be selected) in problem statements, especially with conditions like 'at least' or 'at most'.
⚠️ Common Mistakes
- ✕Confusing combinations with permutations, leading to incorrect problem setup (e.g., using nCr when nPr is required or vice-versa).
- ✕Errors in applying the Addition Principle (for mutually exclusive cases) versus the Multiplication Principle (for sequential events).
- ✕Not correctly handling 'n' and 'r' when conditions are imposed, such as specific items always being included or excluded.
- ✕Algebraic mistakes when solving equations involving nCr or simplifying expressions using its properties.
📝 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