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CBSE · Class 11 · Mathematics

Fundamental Principle of Counting and Factorial (Ch 6: 6.2 and 6.3.2)

3 concepts · 24 practice questions with answers · Free to read

Multiplication principle

From daily life

A shop has 3 shirts and 4 trousers. For each shirt there are 4 trousers to match, so you can dress in 3 × 4 = 12 ways.

Simple definition

If one job can be done in m ways and, after it, a second job can be done in n ways, then the two jobs one after the other can be done in m × n ways. It extends to any number of jobs.

Formula: m × n × p × …

Rule 1: Two jobs — Multiply the ways of each job.

  • Easy: 4 routes from home to school and 3 from school to library: 12 routes.
  • Medium: A code has 1 letter (from 5 letters) and 1 digit (from 4 digits): 20 codes.
  • Exam-type: 3 starters, 4 mains, 2 desserts: 24 meals.

Rule 2: Many jobs — Keep multiplying.

  • Easy: A 3-digit PIN using digits 0 to 9 (repeat allowed): 1000.
  • Medium: A 4-letter code from A, B, C (repeat allowed): 81.
  • Exam-type: A quiz of 5 true/false questions has 32 answer patterns.
Remember

Use multiplication when the jobs are done together (AND).

Common mistake

Adding the ways instead of multiplying.

Memory clue

AND means multiply.

Practice questions

Easy
  1. 13 caps and 5 shirts. How many outfits?
  2. 2A lock has 2 letters from {A, B, C} then 1 digit from 1 to 4 (repeat allowed). How many codes?
  3. 3How many 2-digit PINs are there using digits 0 to 9?
Medium
  1. 4A coin is tossed 4 times. How many outcomes?
  2. 5How many 3-digit numbers can be formed from 1 to 9 if repetition is allowed?
  3. 6How many 3-digit numbers can be formed from 1 to 9 without repetition?
Challenging
  1. 7How many 4-digit numbers exist (first digit not 0)?
  2. 8How many 3-digit even numbers can be made from 1, 2, 3, 4, 5, 6 without repetition?
Show answers (try them yourself first)
  1. 115
  2. 236
  3. 3100
  4. 416
  5. 5729
  6. 6504
  7. 79000
  8. 8Last digit: 3 choices (2, 4, 6); then 5 and 4: 60

Addition principle

From daily life

To reach school you may take a bus (3 routes) or a train (2 routes). You choose one, so there are 3 + 2 = 5 ways.

Simple definition

If one job can be done in m ways and a different job in n ways, and only one of them is to be done, there are m + n ways. The cases must not overlap.

Formula: m + n

Rule 1: Either-or — Add the ways of the separate cases.

  • Easy: 3 pens or 4 pencils to pick one: 7 ways.
  • Medium: Pick a boy (5) or a girl (6) as class captain: 11 ways.
  • Exam-type: Two-digit multiples of 25 are 25, 50, 75: 3 ways.

Rule 2: Both principles — Split into cases (add), and count each case by multiplying.

  • Easy: Passwords of length 1 or 2 from the letters A, B (repeat allowed): 2 + 2 × 2 = 6.
  • Medium: Number of 3-digit numbers that begin with 1 or 2 (any other digits): 200.
  • Exam-type: Number of 2-digit numbers with both digits odd: 25.
Remember

Use addition when the jobs are alternatives (OR).

Common mistake

Using × for "either … or" situations.

Memory clue

OR means add.

Practice questions

Easy
  1. 13 red and 4 blue balls: in how many ways can you pick one ball?
  2. 2A student chooses one subject from 4 science and 3 arts subjects. How many choices?
  3. 3How many 2-digit numbers have both digits even?
Medium
  1. 4How many 2-digit numbers have both digits the same?
  2. 5A bus runs by 4 routes and a train by 3 routes. In how many ways can you go and return by a different mode each time?
  3. 6How many 3-digit numbers have all digits odd?
Challenging
  1. 7How many 3-digit numbers begin with an odd digit and end with an even digit?
  2. 8How many 3-digit numbers have at least one digit equal to 5? (use total − none)
Show answers (try them yourself first)
  1. 17
  2. 27
  3. 3First digit 2,4,6,8 (4), second 0,2,4,6,8 (5): 20
  4. 49
  5. 5Bus there, train back: 4 × 3; train there, bus back: 3 × 4; total 24
  6. 6125
  7. 7250
  8. 8900 − 648 = 252

Factorial

From daily life

Seating 4 friends in a row: 4 choices for the first seat, then 3, then 2, then 1. That product has its own name.

Simple definition

For a natural number n, n! = n × (n − 1) × … × 2 × 1 (read "n factorial"). By definition 0! = 1. Also n! = n × (n − 1)!.

Formula: n! = n × (n − 1)!

Rule 1: Evaluate — Write the product and cancel before multiplying.

  • Easy: 5! = 120.
  • Medium: 7!/5! = 7 × 6 = 42.
  • Exam-type: 8!/(6! × 2!) = 28.

Rule 2: Solve — Break the larger factorial into a smaller one.

  • Easy: n!/(n − 2)! = n(n − 1); if it is 56 then n = 8.
  • Medium: (n + 1)! = 6 × n! gives n + 1 = 6, n = 5.
  • Exam-type: 6!/(4! × 2!) + 4!/(2! × 2!) = 15 + 6 = 21.
Remember

Factorials grow very fast: 10! = 3628800.

Common mistake

Thinking 0! = 0. It is 1.

Memory clue

Write n! as n × (n − 1)! to cancel.

Practice questions

Easy
  1. 1Find 6!.
  2. 2Find 9!/7!.
  3. 3Find 10!/(8! × 2!).
Medium
  1. 4Find 0! + 1! + 2!.
  2. 5If n!/(n − 2)! = 30, find n.
  3. 6Simplify (n + 2)!/n!.
Challenging
  1. 7If (n + 2)! = 12 × n!, find n.
  2. 8Find the value of 1/5! + 1/6! as a single fraction.
Show answers (try them yourself first)
  1. 1720
  2. 272
  3. 345
  4. 44
  5. 5n(n − 1) = 30 gives n = 6
  6. 6(n + 2)(n + 1)
  7. 7(n + 2)(n + 1) = 12 gives n = 2
  8. 8(6 + 1)/6! = 7/720