Selection Problems — Geometry problems (lines, triangles)
Permutation & Combination
9
JEE Qs
8%
Hard
60
min
Thoroughly analyze the geometric constraints (collinearity, parallelism, concurrency) and correctly apply the principles of combinations to subtract invalid formations and count valid ones.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always use combinations (nCr) for selection problems as the order of choosing points to form lines or triangles does not matter.
- 2Carefully identify and subtract 'invalid' cases, especially when points are collinear or lines are parallel, and then add back any valid cases (e.g., the single line formed by collinear points).
- 3Distinguish between selection from a single set of points/lines and selection involving multiple distinct sets (e.g., points on different lines, parallel lines intersecting non-parallel lines).
- 4For intersection problems, explicitly account for conditions like parallel lines, concurrent lines, or lines forming different geometric configurations (e.g., a grid).
⚠️ Common Mistakes
- ✕Forgetting to subtract the invalid combinations formed by collinear points or parallel lines, leading to overcounting.
- ✕Incorrectly applying permutations (nPr) instead of combinations (nCr) when order of selection is irrelevant.
- ✕Overlooking special cases like when all points are collinear, or when multiple sets of parallel lines exist, leading to incomplete counting.
📝 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
- Class 9 Maths Ch 5: Introduction to Euclid's Geometry
- Class 9 Maths Ch 6: Lines and Angles
- Class 9 Maths Ch 7: Triangles