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Number Formation with Distinct Digits: Worksheet 2 - Beginner Practice Number Formation with Distinct Digits BEGINNER

Ready to master Number Formation with Distinct Digits? This entry level practice worksheet (2/10) presents 20 beginner-level challenges. Focus area: pattern recognition. Learn to solve number formation with distinct digits reasoning questions, handle number formation with distinct digits practice, and perfect number formation with distinct digits for competitive exams with our step-by-step solutions.

📝 Worksheet 2 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner level

What you'll learn in this worksheet:
Your progress through Number Formation with Distinct Digits
Worksheet 2 of 10 (11% complete)

Question 1

How many 4-digit numbers with distinct digits are even?
Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

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

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 2 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 2) = 9 × 56 = 504

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 2 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 2) = 1792

Total = 504 + 1792 = 2296

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

Question 2

How many 5-digit numbers with distinct digits are greater than 95902?
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 > 95902:
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 3

How many 5-digit numbers with distinct digits have the digit 4 in position 1 (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 1
- All digits distinct

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


Calculation: 3024

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

Question 4

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

Concept: Fixing a specific digit at a specific position.

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

Step 1 - Handle position 1:
Position 1 (first digit): Must be 3 (1 choice)
- Remaining 2 positions: choose from remaining 9 digits (0-9 except 3) and arrange
- Ways = P(9, 2) = 72


Calculation: 72

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

Question 5

How many 4-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: 4
- 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 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 6

How many 4-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: 4
- 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 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 7

How many 4-digit numbers with distinct digits are divisible by 2?
Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 4 digits
- Digits must be distinct
- Constraint: Divisible by 2

Step 1 - Determine last digit constraint:
Divisible by 2 means:
Last digit must be even

Step 2 - Count valid numbers:
For divisor 2, the count is 3240

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

Calculation: 3240

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

Question 8

How many 4-digit numbers with distinct digits are even?
Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

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

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 2 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 2) = 9 × 56 = 504

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 2 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 2) = 1792

Total = 504 + 1792 = 2296

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

Question 9

How many 4-digit numbers with distinct digits are divisible by 6?
Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 4 digits
- Digits must be distinct
- Constraint: Divisible by 6

Step 1 - Determine last digit constraint:
Divisible by 6 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 6, the count is 756

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

Calculation: 756

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 1 in position 3 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 4
- Digit 1 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 1 (1 choice)
- First digit: cannot be 0 and cannot be 1 (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 3-digit numbers with distinct digits are divisible by 4?
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 4

Step 1 - Determine last digit constraint:
Divisible by 4 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 4, the count is 162

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

Calculation: 162

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

Question 12

How many 3-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: 3 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 72

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

Calculation: 72

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 4-digit numbers with distinct digits are divisible by 4?
Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 4 digits
- Digits must be distinct
- Constraint: Divisible by 4

Step 1 - Determine last digit constraint:
Divisible by 4 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 4, the count is 1134

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

Calculation: 1134

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

Question 14

How many 4-digit numbers with distinct digits are greater than 9254?
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 > 9254:
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 15

How many 4-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: 4 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 504

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

Calculation: 504

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 are divisible by 4?
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 4

Step 1 - Determine last digit constraint:
Divisible by 4 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 4, the count is 162

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

Calculation: 162

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

Question 17

How many 4-digit numbers with distinct digits are divisible by 3?
Step-by-Step Solution:

Concept: Counting numbers with distinct digits and divisibility constraint.

Given:
- Number length: 4 digits
- Digits must be distinct
- Constraint: Divisible by 3

Step 1 - Determine last digit constraint:
Divisible by 3 means:
Special divisibility rules apply

Step 2 - Count valid numbers:
For divisor 3, the count is 1512

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

Calculation: 1512

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

Question 18

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 19

How many 5-digit numbers with distinct digits are greater than 33635?
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 > 33635:
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 20

How many 3-digit numbers with distinct digits are greater than 734?
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 > 734:
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.
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