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Fundamental Counting Principle (FCP)

Introduction

Fundamental Counting Principle (FCP) सभी Permutation और Combination समस्याओं की नींव है। यह हमें यह पता लगाने में मदद करता है कि जब कई स्वतंत्र choices या actions हों, तो कुल कितने possible outcomes बन सकते हैं।

यह pattern इसलिए महत्वपूर्ण है क्योंकि यह हमें बिना हर case को list किए तेज़ी से गिनती करना सिखाता है - जो complex arrangement और selection problems को जल्दी solve करने के लिए एक ज़रूरी skill है।

Pattern: Fundamental Counting Principle (FCP)

Pattern

अगर एक event m तरीकों से हो सकता है और दूसरा n तरीकों से, तो दोनों events एक साथ m × n तरीकों से हो सकते हैं।

दो से ज़्यादा events के लिए, हर एक के तरीकों को multiply करें: Total outcomes = m × n × p × …

Step-by-Step Example

Question

एक restaurant में 3 तरह के starters और 4 तरह के main courses मिलते हैं। एक व्यक्ति कितने तरीकों से एक starter और एक main course चुन सकता है?

Solution

  1. Step 1: दिए गए values पहचानें।

    Number of starters = 3
    Number of main courses = 4

  2. Step 2: Fundamental Counting Principle लागू करें।

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

  3. Step 3: Values substitute करें और calculate करें।

    = 3 × 4 = 12

  4. Final Answer:

    व्यक्ति 12 different ways में meal चुन सकता है।

  5. Quick Check:

    3 starters × 4 mains = 12 total combinations ✅

Quick Variations

1. तीन या उससे ज़्यादा independent choices - जैसे starter, main course, dessert → तीनों को multiply करें।

2. Passwords, license plates या outfit combinations बनाने में उपयोग होता है।

3. कभी-कभी restrictions भी होते हैं (जैसे सिर्फ कुछ digits या letters allowed हों)।

Trick to Always Use

  • Step 1: हर event के options पहचानें।
  • Step 2: अगर choices independent हों, तो सभी possibilities multiply करें।
  • Step 3: Restrictions हों तो उन्हें total combinations के बाद apply करें।

Summary

Summary

Fundamental Counting Principle (FCP) में:

  • Independent choices के outcomes गिनने के लिए multiplication rule का उपयोग करें।
  • Sequential decisions की किसी भी संख्या के लिए rule को extend कर सकते हैं।
  • FCP permutations, combinations और probability calculations की base बनता है।

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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