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Number Formation with Distinct Digits: Worksheet 6 - Intermediate-Advanced Practice Number Formation with Distinct Digits INTERMEDIATE ADVANCED

Ready to master Number Formation with Distinct Digits? This timed practice ⚡ worksheet (6/10) presents 20 intermediate-advanced-level challenges. Focus area: speed building. Learn to solve number formation with distinct digits tricks, handle number formation with distinct digits shortcut methods, and perfect number formation with distinct digits bank exam questions with our step-by-step solutions.

📝 Worksheet 6 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate Advanced level

What you'll learn in this worksheet:
Your progress through Number Formation with Distinct Digits
Worksheet 6 of 10 (55% complete)

Question 1

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 2

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 3

How many 4-digit numbers with distinct digits have the digit 5 in position 2 (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 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (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 4

How many 3-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: 3
- 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 1 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 1) = 8 × 8 = 64


Calculation: 64

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

Question 5

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 6

How many 3-digit numbers with distinct digits have the digit 8 in position 2 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 3
- Digit 8 fixed at position 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (not first): Fixed as 8 (1 choice)
- First digit: cannot be 0 and cannot be 8 (8 choices)
- Remaining 1 positions: choose from remaining 8 digits and arrange
- Ways = 8 × P(8, 1) = 8 × 8 = 64


Calculation: 64

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

Question 7

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

Concept: Fixing a specific digit at a specific position.

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

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 4 (1 choice)
- First digit: cannot be 0 and cannot be 4 (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 8

How many 4-digit numbers with distinct digits are greater than 9352?
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 > 9352:
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 9

How many 4-digit numbers with distinct digits have the digit 6 in position 2 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 4
- Digit 6 fixed at position 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (not first): Fixed as 6 (1 choice)
- First digit: cannot be 0 and cannot be 6 (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 10

How many 4-digit numbers with distinct digits have the digit 9 in position 3 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 4
- Digit 9 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 9 (1 choice)
- First digit: cannot be 0 and cannot be 9 (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 5-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: 5 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 22680

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

Calculation: 22680

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 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 13

How many 5-digit numbers with distinct digits are odd?
Step-by-Step Solution:

Concept: Counting even/odd numbers with distinct digits.

Given: 5-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 3 digits: choose from remaining 8 digits and arrange
- Ways = 5 × 8 × P(8, 3) = 5 × 8 × 336 = 13440

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

Question 14

How many 4-digit numbers with distinct digits have the digit 1 in position 2 (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 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (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 15

How many 4-digit numbers with distinct digits are greater than 9404?
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 > 9404:
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 16

How many 5-digit numbers with distinct digits are greater than 88415?
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 > 88415:
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 17

How many 4-digit numbers with distinct digits are greater than 6038?
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 > 6038:
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 18

How many 5-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: 5 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 6804

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

Calculation: 6804

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

Question 19

How many 5-digit numbers with distinct digits have the digit 2 in position 5 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 5
- Digit 2 fixed at position 5
- All digits distinct

Step 1 - Handle position 5:

Position 5 (not first): Fixed as 2 (1 choice)
- First digit: cannot be 0 and cannot be 2 (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 20

How many 4-digit numbers with distinct digits are greater than 4132?
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 > 4132:
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.
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