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Fundamental Counting Principle (FCP)

Introduction

The Fundamental Counting Principle (FCP) is the foundation of all Permutation and Combination problems. It helps us find the total number of possible outcomes when there are multiple independent choices or actions.

This pattern is important because it allows us to count efficiently without listing every possible case - a crucial skill for solving complex arrangement and selection problems quickly.

Pattern: Fundamental Counting Principle (FCP)

Pattern

If one event can occur in m ways and another in n ways, then both events together can occur in m × n ways.

For more than two events, multiply the number of ways for each: Total outcomes = m × n × p × …

Step-by-Step Example

Question

A restaurant offers 3 types of starters and 4 types of main courses. In how many ways can a person choose one starter and one main course?

Solution

  1. Step 1: Identify what is given.

    Number of starters = 3
    Number of main courses = 4

  2. Step 2: Apply the Fundamental Counting Principle.

    Total possible meal combinations = (Ways to choose starter) × (Ways to choose main course)

  3. Step 3: Substitute and calculate.

    = 3 × 4 = 12

  4. Final Answer:

    The person can choose the meal in 12 different ways.

  5. Quick Check:

    3 starters × 4 mains = 12 total combinations ✅

Quick Variations

1. Three or more independent choices - e.g., starter, main course, dessert → multiply all three.

2. Used in forming passwords, license plates, or outfit combinations.

3. Sometimes involves choices with restrictions (e.g., only certain digits or letters allowed).

Trick to Always Use

  • Step 1: Identify the number of options for each event.
  • Step 2: Multiply all possibilities (if independent).
  • Step 3: Apply restrictions only after finding total combinations.

Summary

Summary

In the Fundamental Counting Principle (FCP):

  • Use multiplication to count outcomes for independent choices.
  • Extend the rule for any number of sequential decisions.
  • FCP forms the base of permutations, combinations, and probability calculations.

Practice

(1/5)
1. A café offers 3 types of coffee and 2 types of pastries. In how many ways can a customer choose one coffee and one pastry?
easy
A. 6
B. 5
C. 8
D. 10

Solution

  1. Step 1: Identify what is given.

    Number of coffee types = 3; number of pastry types = 2.

  2. Step 2: Apply the Fundamental Counting Principle.

    Total ways = (choices for coffee) × (choices for pastry).

  3. Step 3: Substitute and compute.

    3 × 2 = 6.

  4. Final Answer:

    There are 6 possible choices → Option A.

  5. Quick Check:

    Each of 3 coffees pairs with 2 pastries → 3 × 2 = 6 ✅
Hint: Multiply the number of options for each independent choice.
Common Mistakes: Adding counts instead of multiplying the number of options.
2. A password consists of 2 letters followed by 2 digits. If repetition is allowed, how many such passwords can be formed? (Use 26 letters and 10 digits)
easy
A. 67600
B. 676000
C. 6760000
D. 67600000

Solution

  1. Step 1: Identify what is given.

    Each letter position: 26 choices. Each digit position: 10 choices.

  2. Step 2: Apply the Fundamental Counting Principle.

    Total = (choices for letter1) × (letter2) × (digit1) × (digit2).

  3. Step 3: Substitute and compute.

    26 × 26 × 10 × 10 = 26² × 10² = 676 × 100 = 67600.

  4. Final Answer:

    Total passwords = 67600 → Option A.

  5. Quick Check:

    2 letter positions (26²) and 2 digit positions (10²) → 676 × 100 = 67600 ✅
Hint: Treat each position independently and multiply the choices.
Common Mistakes: Forgetting repetition or miscounting positions (exponents).
3. A shop sells 5 brands of pens and 2 brands of pencils. In how many ways can a person buy one pen and one pencil?
easy
A. 7
B. 8
C. 10
D. 12

Solution

  1. Step 1: Identify what is given.

    Number of pen choices = 5; pencil choices = 2.

  2. Step 2: Apply the Fundamental Counting Principle.

    Total ways = (pen choices) × (pencil choices).

  3. Step 3: Substitute and compute.

    5 × 2 = 10.

  4. Final Answer:

    Total possible choices = 10 → Option C.

  5. Quick Check:

    Each of 5 pens pairs with 2 pencils → 5 × 2 = 10 ✅
Hint: For every choice of the first item, multiply by the options for the second.
Common Mistakes: Adding the counts instead of multiplying.
4. A student must choose 1 subject from Maths, Science, English and 1 language from Hindi or French. How many total choices does the student have?
medium
A. 4
B. 5
C. 8
D. 6

Solution

  1. Step 1: Identify what is given.

    Number of subject choices = 3 (Maths, Science, English); number of language choices = 2 (Hindi, French).

  2. Step 2: Apply the Fundamental Counting Principle.

    Total ways = (subject choices) × (language choices).

  3. Step 3: Substitute and compute.

    3 × 2 = 6.

  4. Final Answer:

    The student has 6 possible choices → Option D.

  5. Quick Check:

    3 subjects × 2 languages = 6 ✅
Hint: Multiply number of options across each independent category.
Common Mistakes: Counting subjects and languages separately instead of combining.
5. A car number plate consists of 2 letters followed by 3 digits. If repetition is allowed, how many number plates can be formed? (26 letters and 10 digits available)
medium
A. 67,600
B. 6,76,000
C. 17,57,600
D. 17,57,760

Solution

  1. Step 1: Identify what is given.

    There are 2 letter positions (each 26 choices) and 3 digit positions (each 10 choices).

  2. Step 2: Apply the Fundamental Counting Principle.

    Total plates = 26 × 26 × 10 × 10 × 10.

  3. Step 3: Substitute and compute.

    26² × 10³ = 676 × 1000 = 6,76,000.

  4. Final Answer:

    Total number plates = 6,76,000 → Option B.

  5. Quick Check:

    Two letters → 676 options; three digits → 1000 options; 676 × 1000 = 6,76,000 ✅
Hint: Compute letter combinations and digit combinations separately, then multiply the two results.
Common Mistakes: Forgetting that repetition is allowed or misplacing zeros when multiplying.

Mock Test

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