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

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 5-digit number).

Given:
- Number length: 5 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 4-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 5th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (4 times)
= 9 × 10^4
= 9 × 10000
= 90000

Alternative Verification:
- Smallest 5-digit number: 10000 = 10000
- Largest 5-digit number: 99999 = 99999
- Total count: 99999 - 10000 + 1 = 90000

Related Problems:
1. No repetition: 9 × P(9,4) = 9 × 9!/5!
2. Odd numbers only: 9 × 10^3 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^3 × 1
- If last digit 2,4,6,8: 8 × 10^3 × 4
- Total: 9 × 10^3 + 8 × 4 × 10^3

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 7-digit number).

Given:
- Number length: 7 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 6-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 7th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (6 times)
= 9 × 10^6
= 9 × 1000000
= 9000000

Alternative Verification:
- Smallest 7-digit number: 1000000 = 1000000
- Largest 7-digit number: 9999999 = 9999999
- Total count: 9999999 - 1000000 + 1 = 9000000

Related Problems:
1. No repetition: 9 × P(9,6) = 9 × 9!/3!
2. Odd numbers only: 9 × 10^5 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^5 × 1
- If last digit 2,4,6,8: 8 × 10^5 × 4
- Total: 9 × 10^5 + 8 × 4 × 10^5

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices

Step-by-Step Solution:

Concept: Number formation with positional restrictions. The first digit cannot be 0 (otherwise it wouldn't be an 6-digit number).

Given:
- Number length: 6 digits
- Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
- Repetition: Allowed
- Constraint: First digit cannot be 0

Position-by-Position Analysis:

First digit (leftmost):
Cannot be 0 (would make it 5-digit number)
Choices: 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 9 choices

Second digit:
Can be any digit including 0
Choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Count: 10 choices

Third digit through 6th digit:
Each can be any digit including 0
Count: 10 choices each

Apply Multiplication Principle:
Total numbers = 9 × 10 × 10 × ... × 10 (5 times)
= 9 × 10^5
= 9 × 100000
= 900000

Alternative Verification:
- Smallest 6-digit number: 100000 = 100000
- Largest 6-digit number: 999999 = 999999
- Total count: 999999 - 100000 + 1 = 900000

Related Problems:
1. No repetition: 9 × P(9,5) = 9 × 9!/4!
2. Odd numbers only: 9 × 10^4 × 5 (last digit: 1,3,5,7,9)
3. Even numbers only:
- If last digit 0: 9 × 10^4 × 1
- If last digit 2,4,6,8: 8 × 10^4 × 4
- Total: 9 × 10^4 + 8 × 4 × 10^4

Key Principle: When forming numbers:
- First digit has special restriction (can't be 0)
- Handle positional constraints carefully
- Use multiplication principle for independent choices