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Number Formation with Distinct Digits - Intermediate Level: tricky scenarios handling Number Formation with Distinct Digits INTERMEDIATE

This expert challenge 📈 worksheet focuses on Number Formation with Distinct Digits - a key topic in Permutation Combination. You'll solve 20 intermediate-level problems (Worksheet 5 of 10). The primary focus is on tricky scenarios handling. Master how to solve number formation with distinct digits, number formation with distinct digits tricks, and number formation with distinct digits shortcut methods through systematic practice.

📝 Worksheet 5 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Number Formation with Distinct Digits
Worksheet 5 of 10 (44% complete)

Question 1

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 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 3-digit numbers with distinct digits have the digit 2 in position 3 (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 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 2 (1 choice)
- First digit: cannot be 0 and cannot be 2 (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 4

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 5

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 6

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

Concept: Counting even/odd numbers with distinct digits.

Given: 5-digit numbers, distinct digits, even numbers.

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 3) = 9 × 336 = 3024

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 3) = 10752

Total = 3024 + 10752 = 13776

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 1141?
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 > 1141:
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 4-digit numbers with distinct digits are greater than 5121?
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 > 5121:
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 3-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: 3
- 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 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 10

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

Concept: Counting even/odd numbers with distinct digits.

Given: 5-digit numbers, distinct digits, even numbers.

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 3) = 9 × 336 = 3024

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 3) = 10752

Total = 3024 + 10752 = 13776

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

Question 11

How many 3-digit numbers with distinct digits have the digit 6 in position 3 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 3
- Digit 6 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 6 (1 choice)
- First digit: cannot be 0 and cannot be 6 (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 12

How many 4-digit numbers with distinct digits have the digit 9 in position 4 (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 4
- All digits distinct

Step 1 - Handle position 4:

Position 4 (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 13

How many 3-digit numbers with distinct digits are greater than 370?
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 > 370:
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 14

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

Concept: Counting even/odd numbers with distinct digits.

Given: 5-digit numbers, distinct digits, even numbers.

Case Analysis for even numbers:

Case 1: Last digit = 0
- First digit: 1-9 (9 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 9 × P(8, 3) = 9 × 336 = 3024

Case 2: Last digit = 2,4,6,8 (4 choices)
- First digit: cannot be 0 and cannot be last digit (8 choices)
- Remaining 3 digits: choose from remaining 8 digits and arrange
- Ways = 4 × 8 × P(8, 3) = 10752

Total = 3024 + 10752 = 13776

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

Question 15

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 16

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 17

How many 5-digit numbers with distinct digits are greater than 52605?
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 > 52605:
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 18

How many 3-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: 3 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 216

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

Calculation: 216

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 are greater than 24099?
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 > 24099:
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 4-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: 4
- Digit 4 fixed at position 1
- All digits distinct

Step 1 - Handle position 1:
Position 1 (first digit): Must be 4 (1 choice)
- Remaining 3 positions: choose from remaining 9 digits (0-9 except 4) and arrange
- Ways = P(9, 3) = 504


Calculation: 504

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