Exercise 7.4 Class 11 Math Solutions - Unit 7 Permutations and Combinations, Punjab Board

Step-by-step solutions to Exercise 7.4, the concluding exercise of Unit 7: Permutations and Combinations, for Class 11 Punjab Board students. Covers combination calculations using ⁿCᵣ = n! / [r!(n - r)!], proving algebraic identities (Pascal's rule and complementary combinations), and solving authentic real-world applied problems in cryptography, lottery odds estimation, committee formation, and playlist selections worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 7.4 of Unit 7: Permutations and Combinations for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions on computing combinations using ⁿCᵣ = n!/[r!(n - r)!], solving algebraic equations for n and r, proving algebraic identities such as Pascal's Rule ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ, and analyzing real-world applications in cryptography, lottery odds, committee formations, and song playlist selections are thoroughly worked out for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 7.4 concludes Unit 7: Permutations and Combinations in Class 11 Mathematics by investigating the mathematical theory and real-world utility of Combinations. Unlike permutations where sequence order is paramount, a combination is an unordered selection of r objects chosen from a set of n distinct elements, denoted by ⁿCᵣ = (n choose r) = n! / [r!(n - r)!].

The exercise guides students through fundamental algebraic identities of combinations, including the complementary symmetry ⁿCᵣ = ⁿCₙ₋ᵣ and Pascal's Addition Formula ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. It also features modern, real-world applied combinatorial problems: calculating keyspace complexities in cryptographic password encryption, determining exact odds of winning national lottery drawings, selecting gender-balanced delegatory committees under conditional constraints, and choosing diverse playlist song tracks for specific events.

In Punjab Board HSSC Part 1 examinations, Exercise 7.4 is a major topic for Section B short questions (identity evaluations) and Section C long questions (applied combination word problems) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. Mastering these step-by-step solutions gives students strong conceptual depth and guarantees top scores in board examinations.

FAQs

Are these Exercise 7.4 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering combinations, identity proofs, and real-world applied problems for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 7.4?
Yes, every textbook problem from Exercise 7.4 is completely solved with step-by-step algebraic working, including evaluating ⁿCᵣ, solving equations in n, proving Pascal's rule, and calculating combinatorial possibilities in applied contexts.
What is the difference between a permutation and a combination?
A permutation (ⁿPᵣ) is an arrangement of objects where the order matters, whereas a combination (ⁿCᵣ) is a selection of objects where the order does not matter (ⁿCᵣ = ⁿPᵣ / r!).
What are the complementary combination identity and Pascal's identity?
Complementary combinations state ⁿCᵣ = ⁿCₙ₋ᵣ, and Pascal's Identity states ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.
Can students read these solutions online?
Yes, students can read all Exercise 7.4 solutions directly online through MaryamNotes.pk on any web browser without needing to download external files or install third-party apps.
Are these solutions useful for Punjab Board exams?
Yes, proving combination identities, solving committee selection word problems, and real-world applied combinatorics are frequent high-yield questions in Section B short questions and Section C long questions.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are fully aligned with the updated Student Learning Outcomes (SLO) mandated by the Punjab Curriculum and Textbook Board (PECTAA) for the 2026-2027 academic year.
Are these solutions free for students?
Yes, all Class 11 Mathematics Exercise 7.4 solutions on MaryamNotes.pk are available entirely free of cost for students and educators across Punjab.