1st Year Math Chapter 7 Exercise 7.2 Notes: Permutation (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 7.2 introduces permutations — arrangements of objects where the order matters — building directly on the counting principles and factorial notation from Exercise 7.1.
What Does This Exercise Cover?
This exercise teaches the permutation formula for arranging r objects chosen from a group of n distinct objects, and how to apply it to arrangement problems.
Key Concepts
1. What Is a Permutation?
A permutation is an arrangement of objects in a specific order. Changing the order of the same objects produces a different permutation.
2. Permutation of n Objects Taken r at a Time
The number of ways to arrange r objects chosen from n distinct objects, denoted nPr, is given by nPr = n! / (n − r)!.
3. Arranging All n Objects at Once
When all n distinct objects are arranged (r = n), the formula simplifies to nPn = n!, since (n − n)! = 0! = 1.
Solved Examples
Example 1: Evaluate 6P2.
6P2 = 6! / (6 − 2)! = 6! / 4! = 720 / 24.
Answer: 30
Example 2: In how many ways can 3 books be selected and arranged from 5 different books on a shelf?
This is 5P3 = 5! / (5 − 3)! = 120 / 2.
Answer: 60 ways
Example 3: In how many ways can 4 people be arranged in a row?
This is 4P4 = 4! = 4 × 3 × 2 × 1.
Answer: 24 ways
Sample MCQs
1. A permutation is:
a) A selection where order doesn’t matter b) An arrangement where order matters c) Always a combination d) Undefined for repeated objects
Answer: b) An arrangement where order matters
2. The formula for nPr is:
a) n! / (n − r)! b) n! / (r!(n − r)!) c) n! × r! d) n! / r!
Answer: a) n! / (n − r)!
3. The value of 5P2 is:
a) 10 b) 15 c) 20 d) 25
Answer: c) 20
4. nPn equals:
a) n b) n! c) 1 d) 0
Answer: b) n!
5. The number of ways to arrange 4 distinct books on a shelf is:
a) 4 b) 12 c) 16 d) 24
Answer: d) 24
Important Short Questions
- Define permutation.
- State the formula for nPr.
- Evaluate 7P3.
- How many ways can 5 people be arranged in a row?
- What is the value of nPn?
Important Long Questions
- In how many ways can 4 letters be selected and arranged from the 6 distinct letters of the word NUMBER?
- Evaluate 8P5, showing each step of the calculation.
- In how many ways can first, second, and third prizes be awarded among 10 contestants, assuming no ties?
- Explain the difference between a permutation and simply selecting objects (without arranging them), using an example.
How to Approach This Exercise Effectively
- Always ask ‘does order matter here?’ first — if yes, you’re dealing with a permutation.
- Write out the nPr formula in full before substituting numbers, to avoid mixing up n and r.
- For ‘arrange all objects’ problems, remember the shortcut nPn = n! directly, without needing the full formula.
- Practice canceling factorials algebraically (like 8!/5! = 8×7×6) instead of computing both factorials fully — it’s much faster.
- Watch for wording like ‘first, second, third place’ — this is a strong signal that order matters and permutations apply.
FAQs
Q: What’s the difference between nPr and n!?
A: n! arranges all n objects, while nPr arranges only r of the n objects — nPr becomes n! specifically when r equals n.
Q: Does order matter in a permutation?
A: Yes — that’s the defining feature of a permutation; the same objects in a different order count as a different permutation.
Q: Can r be greater than n in nPr?
A: No — you cannot arrange more objects than you have available, so r must be less than or equal to n.
Q: Why is the formula n!/(n−r)! used for permutations?
A: It counts all possible orderings of n objects, then divides out the orderings of the (n−r) objects that weren’t chosen, since their internal order doesn’t matter.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 7: Permutations and Combinations, Exercise 7.2.