0
0

All Letters Different (Word Arrangement)

Introduction

When all letters (or items) are distinct, arranging them is straightforward - every order is unique. This pattern appears in word arrangement, ordering objects, or listing distinct items.

It is important because many exam questions ask for permutations of distinct objects; recognizing that all items are different lets you use the factorial rule directly.

Pattern: All Letters Different (Word Arrangement)

Pattern

The key idea: if you have n distinct letters and you arrange all of them, the number of arrangements is n!

Formula:
Total arrangements = n!

Why this works: For the first position there are n choices, for the second n-1, continuing until 1 → multiply them: n × (n - 1) × ... × 1 = n!.

Step-by-Step Example

Question

How many distinct arrangements (anagrams) can be made from the letters of the word PLANT?

Solution

  1. Step 1: Identify what is given.

    The word PLANT has n = 5 letters and all letters are distinct.

  2. Step 2: Choose the formula.

    Use n! = 5! because all items are different and we arrange all of them.

  3. Step 3: Compute.

    5! = 5 × 4 × 3 × 2 × 1 = 120

  4. Final Answer:

    There are 120 distinct arrangements of the letters of PLANT.

  5. Quick Check:

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

Quick Variations

1. Partial arrangement: If you arrange only r of the n distinct letters, use nPr = n! / (n - r)!.

2. Distinct positions: If positions have labels (e.g., seat numbers), treat them as ordered and use factorial/permutation rules.

3. Mixed types: If some letters repeat, switch to the repeated-letters formula (divide by factorials of repeats).

Trick to Always Use

  • Step 1 → Check: are all items distinct? If YES → use n!.
  • Step 2 → For arranging all items, compute n! directly; for arranging only some, use nPr (multiply top r factors).

Summary

Summary

Key takeaways:

  • If all n letters are different and you arrange all of them, total arrangements = n!.
  • Compute factorial by multiplying descending integers (n × (n - 1) × ... × 1).
  • For partial arrangements or repeated items, use the appropriate permutation or repeated-letter formulas instead.

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

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes