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All Letters Different (Word Arrangement)

Introduction

जब सभी letters (या items) अलग-अलग हों, तो उन्हें arrange करना सीधा होता है - हर order unique होता है। यह pattern word arrangement, objects को order करने या distinct items की listing में दिखाई देता है।

यह महत्वपूर्ण है क्योंकि कई exam questions distinct objects की permutations पूछते हैं; सभी items अलग हों तो आप सीधे factorial rule use कर सकते हैं।

Pattern: All Letters Different (Word Arrangement)

Pattern

मुख्य idea: अगर आपके पास n distinct letters हों और आप सभी को arrange करें, तो total arrangements = n!

Formula:
Total arrangements = n!

Why this works: पहली position के लिए n choices, दूसरी के लिए n-1, ऐसे ही आखिरी तक 1 → multiply करें: n × (n - 1) × ... × 1 = n!.

Step-by-Step Example

Question

PLANT शब्द की letters से कितने distinct arrangements (anagrams) बनाए जा सकते हैं?

Solution

  1. Step 1: दिए गए values पहचानें।

    शब्द PLANT में n = 5 letters हैं और सभी distinct हैं।

  2. Step 2: Formula चुनें।

    Use n! = 5! क्योंकि सभी items अलग हैं और सभी को arrange कर रहे हैं।

  3. Step 3: Compute करें।

    5! = 5 × 4 × 3 × 2 × 1 = 120

  4. Final Answer:

    PLANT की letters से 120 distinct arrangements बनते हैं।

  5. Quick Check:

    Positions: 1st = 5 choices, 2nd = 4, 3rd = 3, 4th = 2, 5th = 1 → multiply = 120 ✅

Quick Variations

1. Partial arrangement: केवल r letters arrange करने हों → nPr = n! / (n - r)!.

2. Distinct positions: अगर positions को labels दिए हों (जैसे seat numbers), तो उन्हें ordered मानकर factorial/permutation rules use करें।

3. Mixed types: अगर कुछ letters repeat हों, तो repeated-letters formula (repeats के factorial से divide) use करें।

Trick to Always Use

  • Step 1 → Check: क्या सभी items distinct हैं? अगर YES → n!।
  • Step 2 → सभी items arrange करने पर n! directly compute करें; कुछ items arrange करने पर nPr (top r factors multiply) use करें।

Summary

Summary

Key takeaways:

  • अगर सभी n letters अलग हों और आप सभी को arrange करें, total arrangements = n!.
  • Factorial = descending integers multiply करना (n × (n - 1) × ... × 1).
  • Partial arrangements या repeated items होने पर सही permutation/repeated-letter formula चुनें।

Practice

(1/5)
1. How many different words can be formed using all the letters of the word ‘CHAIR’?
easy
A. 60
B. 100
C. 120
D. 80

Solution

  1. Step 1: Identify given data.

    The word 'CHAIR' has 5 distinct letters.

  2. Step 2: Choose the formula.

    All letters different and all used → total arrangements = n! where n = 5.

  3. Step 3: Compute.

    5! = 5 × 4 × 3 × 2 × 1 = 120.

  4. Final Answer:

    There are 120 different words → Option C.

  5. Quick Check:

    Positions: 5 × 4 × 3 × 2 × 1 = 120 ✅
Hint: If all letters are distinct and all are used, compute n! directly.
Common Mistakes: Using combinations or partial-arrangement formulas instead of factorial.
2. Find the number of different 6-letter arrangements possible with the word ‘SILVER’.
easy
A. 720
B. 240
C. 360
D. 120

Solution

  1. Step 1: Identify given data.

    The word 'SILVER' has 6 distinct letters.

  2. Step 2: Choose the formula.

    All letters different and all are used → total = 6!.

  3. Step 3: Compute.

    6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

  4. Final Answer:

    There are 720 arrangements → Option A.

  5. Quick Check:

    Multiply descending counts: 6 × 5 × 4 × 3 × 2 × 1 = 720 ✅
Hint: Count letters, confirm all distinct, then use n! for arrangements.
Common Mistakes: Forgetting to check for repeated letters before using n!.
3. How many distinct 4-letter words can be made from the letters of the word ‘PLANT’ (no repetition)?
easy
A. 120
B. 24
C. 60
D. 48

Solution

  1. Step 1: Identify given data.

    'PLANT' has 5 distinct letters; we need ordered 4-letter words (r = 4).

  2. Step 2: Choose the formula.

    Arrange r out of n distinct letters → nPr = n! / (n - r)! with n = 5, r = 4.

  3. Step 3: Compute.

    5P4 = 5! / 1! = 5 × 4 × 3 × 2 = 120.

  4. Final Answer:

    There are 120 distinct 4-letter words → Option A.

  5. Quick Check:

    Choose positions sequentially: 5 × 4 × 3 × 2 = 120 ✅
Hint: When arranging r out of n distinct items, multiply top r factors: n × (n - 1) × ... for r terms.
Common Mistakes: Using n! when not all letters are used or using combinations instead of permutations.
4. How many different arrangements can be made using all the letters of the word ‘GARDEN’?
medium
A. 360
B. 720
C. 840
D. 120

Solution

  1. Step 1: Identify given data.

    The word 'GARDEN' has 6 distinct letters.

  2. Step 2: Choose the formula.

    All letters are used and distinct → total arrangements = 6!.

  3. Step 3: Compute.

    6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

  4. Final Answer:

    There are 720 arrangements → Option B.

  5. Quick Check:

    6! = 720 ✅
Hint: Confirm letters are distinct; then use factorial for full arrangements.
Common Mistakes: Assuming repeated letters when none exist or using partial-arrangement formulas incorrectly.
5. How many ordered 4-letter arrangements can be formed from 6 distinct letters?
medium
A. 240
B. 360
C. 720
D. 168

Solution

  1. Step 1: Identify given data.

    n = 6 distinct letters; r = 4 positions; order matters, no repetition.

  2. Step 2: Choose the formula.

    nPr = n! / (n - r)!. Here 6P4 = 6! / 2!.

  3. Step 3: Compute.

    6P4 = 6 × 5 × 4 × 3 = 360.

  4. Final Answer:

    There are 360 ordered arrangements → Option B.

  5. Quick Check:

    Pick positions sequentially: 6 × 5 × 4 × 3 = 360 ✅
Hint: For arranging r out of n distinct items, multiply the top r descending factors of n!.
Common Mistakes: Using combinations (which ignore order) instead of permutations for ordered arrangements.

Mock Test

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