Exercise 7.3 Class 11 Math Solutions - Unit 7 Permutations and Combinations, Punjab Board
Step-by-step solutions to Exercise 7.3, Unit 7: Permutations and Combinations, for Class 11 Punjab Board students. Covers permutations of multisets where objects are not all distinct n! / (n₁! n₂! ... nₖ!), and circular permutations for both distinct orientations (n - 1)! and identical clockwise/anticlockwise reflections (n - 1)! / 2 worked out question by question.
Exercise Overview
Exercise 7.3 in Unit 7 of Class 11 Mathematics extends the concept of permutations to two specialized domains: multisets containing repeating identical items, and non-linear circular arrangements. When arranging n total objects where n₁ are identical of type 1, n₂ of type 2, and so on, interchanging identical items produces indistinguishable configurations. The total number of unique linear permutations is therefore reduced to n! / (n₁! · n₂! ··· nₖ!).
The exercise also examines Circular Permutations, where shifting all elements by one position along a loop leaves the relative cyclic ordering unchanged. By fixing one reference element, the number of circular permutations of n distinct items is (n - 1)!. Students explore two distinct physical conditions: Case 1, where clockwise and anticlockwise perspectives are distinguishable (e.g., people seated around a dining table), yielding (n - 1)! arrangements; and Case 2, where reversing or flipping the arrangement in three dimensions makes clockwise and anticlockwise orientations identical (e.g., threading beads on a necklace or keys on a ring), dividing the total by 2 to yield (n - 1)! / 2.
In Punjab Board HSSC Part 1 examinations, Exercise 7.3 problems are classic staples for Section B short questions (word permutations such as 'STATISTICS') and Section C long questions (circular seating and necklace proofs) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. Practicing these step-by-step solutions gives students mastery over symmetry reduction and multiset combinatorics.