Permutations and Combinations | General Aptitude for GATE PSU ESE AE/JE

Permutations and Combinations (P&C) is an important topic in General Aptitude and is frequently asked in exams like GATE, PSU, ESE, AE/JE. It helps in solving problems related to arrangements, selections, and counting techniques.

Basic Difference

• Permutation: Arrangement of objects where order matters
• Combination: Selection of objects where order does not matter

Factorial Concept

n! = n × (n − 1) × (n − 2) × ... × 1

0! = 1

Example:

5! = 5 × 4 × 3 × 2 × 1 = 120

Permutation Formula

nPr = n! / (n − r)!

Example:

Number of ways to arrange 3 items out of 5:
5P3 = 5! / 2! = 60

Combination Formula

nCr = n! / [r! (n − r)!]

Example:

Number of ways to select 3 items out of 5:
5C3 = 5! / (3! × 2!) = 10

Important Properties

• nCr = nC(n − r)
• nCr + nC(r − 1) = (n + 1)Cr
• nPr = nCr × r!

Important Tricks

• Use symmetry: nCr = nC(n−r)
• Simplify factorials before calculation
• Memorize common values (like 5C2, 6C3, etc.)

GATE Level Questions

Question 1

In how many ways can 4 students be selected from 10 students?

Solution:
10C4 = 10! / (4! × 6!) = 210
Answer: 210

Question 2

In how many ways can 3 books be arranged from 6 different books?

Solution:
6P3 = 6! / 3! = 120
Answer: 120

Question 3

Find the value of 8C2.

Solution:
8C2 = (8 × 7) / 2 = 28
Answer: 28

Common Mistakes to Avoid

• Confusing permutation with combination
• Not simplifying factorials properly
• Calculation errors in large numbers

Exam Tips

• Identify whether order matters or not
• Practice shortcut simplifications
• Solve previous year questions regularly

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