Factorials & the Counting Principle

🎯 Learning Objective

Use factorials, the fundamental counting principle and nPr to count arrangements of letters and digits.

💡 Concept

• n! = n × (n−1) × … × 1, and 0! = 1 by definition
• Values to memorise: 3! = 6, 4! = 24, 5! = 120, 6! = 720
• Fundamental counting principle: first task m ways AND second task n ways → m × n total ways
• nPr = n!/(n−r)! counts ARRANGEMENTS — order matters
• n distinct objects in a row → n! ways; repeated letters divide: arrangements = n!/(p! q!)

🧮 Key Formulas

n! = n × (n−1) × … × 1; 0! = 1

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nPr = n!/(n−r)!

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Word with repeats: n!/(p! q!)

✏️ Easy Example

Q. How many 3-digit numbers can be formed using the digits 1–5 without repetition?

1. First place: 5 choices
2. Second: 4, Third: 3
3. 5 × 4 × 3

Answer: 60

🇮🇳 Real-Life Example

A scooter's 4-digit number plate part: 10 × 10 × 10 × 10 = 10,000 possibilities — the counting principle runs every RTO office.

📝 Exam-Level Example

Q. In how many ways can the letters of the word TRAIN be arranged?

1. 5 distinct letters
2. Arrangements = 5!
3. = 5 × 4 × 3 × 2 × 1

Answer: 120

📝 Exam-Level Example

Q. In how many ways can the letters of the word APPLE be arranged?

1. 5 letters, P repeats twice
2. = 5!/2!
3. = 120 ÷ 2

Answer: 60

🪄 Memory Trick

Prize questions need no formula: gold and silver among 8 runners = 8 × 7 = 56. Just multiply falling numbers, one per position.

⚠️ Common Mistakes

• ❌ Multiplying when tasks are alternatives — OR means add, AND means multiply
• ❌ Forgetting to divide by the factorial of repeated letters
• ❌ Taking 0! = 0 — it is 1

🏆 Exam Tips

• ✅ Draw blanks ( _ _ _ ) and fill the number of choices per position
• ✅ Number-forming questions: the first digit cannot be 0 — fix it first

📌 Summary

• n! counts arrangements of n distinct things
• AND → multiply, OR → add
• nPr = n!/(n−r)! when order matters
• Repeated letters → divide by their factorials