Maths · Algebra · chapter 6 of 14

Class 11 Maths Chapter 6: Permutations and Combinations case study question with answers

One case-based question of the kind CBSE-pattern school exams ask: a passage and a figure, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.

Case study · 4 marks
Ananya's home H and her school S are corners of a grid of streets, as shown. She always walks by a shortest route: every block she walks is either one block east (E) or one block north (N). The school is 5 blocks east and 3 blocks north of her home, so a route such as EENENENE is written as a word of 5 Es and 3 Ns. Her favourite sweet shop is at the corner K, 2 blocks east and 1 block north of H.
H (home)S (school)KE →N ↑
Part 1 · 1 mark
Which of the following words is not a shortest route from H to S?
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Answer: (c) EENENNEN

A shortest route has exactly 5 Es and 3 Ns. EENENNEN has 4 Es and 4 Ns, so it ends 4 blocks east and 4 blocks north of H, not at S; each of the other three words has 5 Es and 3 Ns.

Part 2 · 1 mark
The number of shortest routes from H to S is
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Answer: (c) 56

8 letters, 5 alike of one kind and 3 of another: 8!/(5! 3!) = 56.

Part 3 · 2 marks
How many shortest routes from H to S pass through the sweet shop K?
Answer and marking scheme

H to K: 2 Es and 1 N, 3!/(2! 1!) = 3 routes. K to S: 3 Es and 2 Ns, 5!/(3! 2!) = 10 routes. By the multiplication principle, 3 × 10 = 30.

  • H to K: 3 routes and K to S: 10 routes (1)
  • multiplies: 30 (1)
Or
How many shortest routes from H to S avoid the sweet shop K?
Answer and marking scheme

Routes through K: H to K, 3!/2! = 3; K to S, 5!/(3! 2!) = 10; so 3 × 10 = 30. Routes avoiding K: 56 − 30 = 26.

  • routes through K: 3 × 10 = 30 (1)
  • 56 − 30 = 26 (1)
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