Number Formation with Distinct Digits Beginner-Intermediate Worksheet: Focus on common variations practice Number Formation with Distinct Digits BEGINNER INTERMEDIATE

Level up your Number Formation with Distinct Digits skills! You're at Worksheet 4 of 10 (33% through this series). This step-up challenge worksheet features 20 beginner-intermediate-level problems with a focus on common variations practice. Topics covered: number formation with distinct digits for competitive exams, how to solve number formation with distinct digits, number formation with distinct digits tricks.

📝 Worksheet 4 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner Intermediate level

What you'll learn in this worksheet:

Understand the logic behind how to solve number formation with distinct digits
Learn step-by-step approaches to common variations practice
Bridge the gap between basic and advanced concepts
Handle problems with increasing complexity
Master number formation with distinct digits for competitive exams through focused practice

Your progress through Number Formation with Distinct Digits

Worksheet 4 of 10 (33% complete)

Question 1

How many 4-digit numbers with distinct digits have the digit 5 in position 3 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 4
- Digit 5 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 5 (1 choice)
- First digit: cannot be 0 and cannot be 5 (8 choices)
- Remaining 2 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 2) = 8 × 56 = 448

Calculation: 448

Key Point: When fixing a digit in first position, it cannot be 0.

Question 2

How many 5-digit numbers with distinct digits have the digit 6 in position 1 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 5
- Digit 6 fixed at position 1
- All digits distinct

Step 1 - Handle position 1:
Position 1 (first digit): Must be 6 (1 choice)
- Remaining 4 positions: choose from remaining 9 digits (0-9 except 6) and arrange
- Ways = P(9, 4) = 3024

Calculation: 3024

Key Point: When fixing a digit in first position, it cannot be 0.

Question 3

How many 3-digit numbers with distinct digits are greater than 756?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 2 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 2) = 9 × 72 = 648

Step 2 - Count those > 756:
Approximately half of all numbers will be greater than the median.
Answer ≈ 324

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 4

How many 3-digit numbers with distinct digits are greater than 400?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 2 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 2) = 9 × 72 = 648

Step 2 - Count those > 400:
Approximately half of all numbers will be greater than the median.
Answer ≈ 324

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 5

How many 3-digit numbers with distinct digits are odd?

Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

Given: 3-digit numbers, distinct digits, odd numbers.

Case Analysis for odd numbers:

- Last digit: 1,3,5,7,9 (5 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 1 digits: choose from remaining 8 digits and arrange
- Ways = 5 × 8 × P(8, 1) = 5 × 8 × 8 = 320

Key Principle: Always handle first digit (can't be 0) and last digit (parity constraint) separately.

Question 6

How many 3-digit numbers with distinct digits are even?

Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

Given: 3-digit numbers, distinct digits, even numbers.

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 1 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 1) = 9 × 8 = 72

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 1 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 1) = 256

Total = 72 + 256 = 328

Key Principle: Always handle first digit (can't be 0) and last digit (parity constraint) separately.

Question 7

How many 4-digit numbers with distinct digits are greater than 9530?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 3 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 3) = 9 × 504 = 4536

Step 2 - Count those > 9530:
Approximately half of all numbers will be greater than the median.
Answer ≈ 2268

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 8

How many 3-digit numbers with distinct digits are greater than 343?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 2 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 2) = 9 × 72 = 648

Step 2 - Count those > 343:
Approximately half of all numbers will be greater than the median.
Answer ≈ 324

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 9

How many 5-digit numbers with distinct digits are divisible by 10?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 5 digits
- Digits must be distinct
- Constraint: Divisible by 10

Step 1 - Determine last digit constraint:
Divisible by 10 means:
Last digit must be 0

Step 2 - Count valid numbers:
For divisor 10, the count is 4536

Step 3 - Verify distinctness:
All digits in the number are different (no repetition allowed).

Calculation: 4536

Key Principle: When forming numbers with distinct digits:
- First digit cannot be 0
- Handle last digit constraints first
- Use permutations for remaining positions

Question 10

How many 4-digit numbers with distinct digits have the digit 3 in position 2 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 4
- Digit 3 fixed at position 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (not first): Fixed as 3 (1 choice)
- First digit: cannot be 0 and cannot be 3 (8 choices)
- Remaining 2 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 2) = 8 × 56 = 448

Calculation: 448

Key Point: When fixing a digit in first position, it cannot be 0.

Question 11

How many 4-digit numbers with distinct digits are greater than 8331?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 3 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 3) = 9 × 504 = 4536

Step 2 - Count those > 8331:
Approximately half of all numbers will be greater than the median.
Answer ≈ 2268

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 12

How many 3-digit numbers with distinct digits are divisible by 10?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 3 digits
- Digits must be distinct
- Constraint: Divisible by 10

Step 1 - Determine last digit constraint:
Divisible by 10 means:
Last digit must be 0

Step 2 - Count valid numbers:
For divisor 10, the count is 81

Step 3 - Verify distinctness:
All digits in the number are different (no repetition allowed).

Calculation: 81

Key Principle: When forming numbers with distinct digits:
- First digit cannot be 0
- Handle last digit constraints first
- Use permutations for remaining positions

Question 13

How many 5-digit numbers with distinct digits are greater than 45380?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits greater than a threshold.

Step 1 - Total distinct-digit numbers:
First digit: 1-9 (9 choices)
Remaining 4 digits: choose and arrange from remaining 9 digits
Total = 9 × P(9, 4) = 9 × 3024 = 27216

Step 2 - Count those > 45380:
Approximately half of all numbers will be greater than the median.
Answer ≈ 13608

Note: Exact calculation would require case-by-case analysis based on the first few digits.

Question 14

How many 5-digit numbers with distinct digits have the digit 8 in position 3 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 5
- Digit 8 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 8 (1 choice)
- First digit: cannot be 0 and cannot be 8 (8 choices)
- Remaining 3 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 3) = 8 × 336 = 2688

Calculation: 2688

Key Point: When fixing a digit in first position, it cannot be 0.

Question 15

How many 5-digit numbers with distinct digits are divisible by 9?

Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 5 digits
- Digits must be distinct
- Constraint: Divisible by 9

Step 1 - Determine last digit constraint:
Divisible by 9 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 9, the count is 3024

Step 3 - Verify distinctness:
All digits in the number are different (no repetition allowed).

Calculation: 3024

Key Principle: When forming numbers with distinct digits:
- First digit cannot be 0
- Handle last digit constraints first
- Use permutations for remaining positions

Question 16

How many 3-digit numbers with distinct digits have the digit 2 in position 1 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 3
- Digit 2 fixed at position 1
- All digits distinct

Step 1 - Handle position 1:
Position 1 (first digit): Must be 2 (1 choice)
- Remaining 2 positions: choose from remaining 9 digits (0-9 except 2) and arrange
- Ways = P(9, 2) = 72

Calculation: 72

Key Point: When fixing a digit in first position, it cannot be 0.

Question 17

How many 5-digit numbers with distinct digits have the digit 4 in position 5 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 5
- Digit 4 fixed at position 5
- All digits distinct

Step 1 - Handle position 5:

Position 5 (not first): Fixed as 4 (1 choice)
- First digit: cannot be 0 and cannot be 4 (8 choices)
- Remaining 3 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 3) = 8 × 336 = 2688

Calculation: 2688

Key Point: When fixing a digit in first position, it cannot be 0.

Question 18

How many 3-digit numbers with distinct digits are odd?

Question 19

How many 4-digit numbers with distinct digits are odd?

Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

Given: 4-digit numbers, distinct digits, odd numbers.

Case Analysis for odd numbers:

- Last digit: 1,3,5,7,9 (5 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 2 digits: choose from remaining 8 digits and arrange
- Ways = 5 × 8 × P(8, 2) = 5 × 8 × 56 = 2240

Key Principle: Always handle first digit (can't be 0) and last digit (parity constraint) separately.

Question 20

How many 5-digit numbers with distinct digits have the digit 7 in position 4 (counting from left)?

Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 5
- Digit 7 fixed at position 4
- All digits distinct

Step 1 - Handle position 4:

Position 4 (not first): Fixed as 7 (1 choice)
- First digit: cannot be 0 and cannot be 7 (8 choices)
- Remaining 3 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 3) = 8 × 336 = 2688

Calculation: 2688

Key Point: When fixing a digit in first position, it cannot be 0.

📝 Continue your Number Formation with Distinct Digits practice. Worksheet 4 focuses on common variations practice.

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