Derangement
Permutation & Combination
9
JEE Qs
8%
Hard
45
min
Master the derivation using the Principle of Inclusion-Exclusion and memorize the direct formula along with the first few values to tackle advanced P&C problems efficiently.
🧮 Key Formulas
✅ Key Points for JEE
- 1Derangement refers to an arrangement of 'n' distinct objects such that none of them occupy their original (or designated) positions.
- 2The formula for D_n is fundamentally derived using the Principle of Inclusion-Exclusion.
- 3Memorize the values for small 'n': D_0=1, D_1=0, D_2=1, D_3=2, D_4=9, D_5=44, as they frequently appear in problems.
- 4The recursive relation D_n = (n-1)(D_{n-1} + D_{n-2}) is particularly useful for quickly calculating D_n values or solving problems involving related recurrence relations.
⚠️ Common Mistakes
- ✕Confusing derangements with permutations where 'some' objects are in their original position, instead of 'none'.
- ✕Errors in calculation, especially with alternating signs or factorials in the main formula.
- ✕Not remembering the base cases for the recursive formula (D_0=1, D_1=0), leading to incorrect iterative calculations.
📝 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