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MathsMediumClass 11

Permutations ā€” nPr, with repetition, circular

Permutation & Combination

9

JEE Qs

8%

Hard

75

min

Always clearly identify whether 'order matters' and if 'repetitions are allowed/present' to select the correct permutation method for a given problem.

šŸ§® Key Formulas

P(n, r) = n! / (n-r)! (Permutations of n distinct items taken r at a time)
Permutations of n items where pā‚ are of one kind, pā‚‚ of another, ..., pā‚– of another: n! / (pā‚! pā‚‚! ... pā‚–!)
Circular permutations of n distinct items: (n-1)!
Circular permutations of n distinct items when clockwise and anti-clockwise arrangements are considered identical (e.g., beads in a necklace): (n-1)! / 2

āœ… Key Points for JEE

  • 1Permutations specifically deal with arrangements where the order of items is significant. If order doesn't matter, it's a combination problem.
  • 2The Fundamental Principle of Counting (Multiplication Rule) is the foundational concept for solving permutation problems, with nPr being a specialized formula for distinct items.
  • 3When identical items are present, divide the total number of arrangements by the factorial of the count of each type of identical item to correct for overcounting.
  • 4For circular permutations, fix one item's position to eliminate rotational symmetry and convert it into a linear arrangement problem, leading to (n-1)! for distinct items.
  • 5Carefully distinguish between circular arrangements where only relative order matters (like seating arrangements) and those where reflection symmetry makes arrangements indistinguishable (like forming a necklace).

āš ļø Common Mistakes

  • āœ•Confusing permutations with combinations; incorrectly using nPr when nCr is required or vice-versa.
  • āœ•Failing to account for repetitions by not dividing by the factorials of identical items, or dividing by incorrect factors.
  • āœ•Not properly handling conditions or restrictions (e.g., specific items must be together, specific positions must be filled).
  • āœ•Overlooking the special case of circular permutations where clockwise and anti-clockwise arrangements are identical, leading to forgetting the division by 2.

šŸ“ Practice Questions

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Q2. 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

2025Ā·MCQMedium

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 āˆ’10x + 9 = 0 (2) x2 + y2 āˆ’6x + 5 = 0 (3) x2 + y2 āˆ’4x + 3 = 0 (4) x2 + y2 āˆ’8x + 7 = 0

2025Ā·MCQMedium

Q25.The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is______.

2025Ā·NumericalMedium

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

2025Ā·MCQMedium

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

2025Ā·MCQMedium

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 -

2025Ā·NumericalMedium

NCERT Chapters

  • Class 11 Mathematics Ch 7: Permutations and Combinations