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Number Formation with Distinct Digits - Expert Level: conceptual clarity Number Formation with Distinct Digits EXPERT

This skill evaluation ⚡ worksheet focuses on Number Formation with Distinct Digits - a key topic in Permutation Combination. You'll solve 20 expert-level problems (Worksheet 9 of 10). The primary focus is on conceptual clarity. Master number formation with distinct digits ssc cgl, number formation with distinct digits reasoning tricks, and fast number formation with distinct digits solving through systematic practice.

📝 Worksheet 9 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Expert level

What you'll learn in this worksheet:
Your progress through Number Formation with Distinct Digits
Worksheet 9 of 10 (88% complete)

Question 1

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

Step 1 - Handle position 4:

Position 4 (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 2

How many 3-digit numbers with distinct digits are greater than 577?
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 > 577:
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 3

How many 3-digit numbers with distinct digits have the digit 7 in position 3 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 3
- Digit 7 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 7 (1 choice)
- First digit: cannot be 0 and cannot be 7 (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 3?
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 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 9072

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

Calculation: 9072

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 3-digit numbers with distinct digits are greater than 245?
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 > 245:
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 6

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 7

How many 3-digit numbers with distinct digits are greater than 391?
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 > 391:
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 8

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

Step 1 - Determine last digit constraint:
Divisible by 5 means:
Last digit must be 0 or 5

Step 2 - Count valid numbers:
For divisor 5, the count is 5712

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

Calculation: 5712

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

Question 9

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

Step 1 - Handle position 4:

Position 4 (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 10

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

Step 1 - Handle position 4:

Position 4 (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 11

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 12

How many 3-digit numbers with distinct digits are greater than 328?
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 > 328:
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 13

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 14

How many 3-digit numbers with distinct digits have the digit 7 in position 3 (counting from left)?
Step-by-Step Solution:

Concept: Fixing a specific digit at a specific position.

Given:
- Number length: 3
- Digit 7 fixed at position 3
- All digits distinct

Step 1 - Handle position 3:

Position 3 (not first): Fixed as 7 (1 choice)
- First digit: cannot be 0 and cannot be 7 (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 15

How many 4-digit numbers with distinct digits have the digit 7 in position 2 (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 2
- All digits distinct

Step 1 - Handle position 2:

Position 2 (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 16

How many 4-digit numbers with distinct digits are greater than 6665?
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 > 6665:
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 17

How many 4-digit numbers with distinct digits are greater than 5282?
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 > 5282:
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 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 19

How many 4-digit numbers with distinct digits are greater than 5084?
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 > 5084:
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 20

How many 4-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: 4
- 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 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.
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