How many 4-digit numbers can be generated from the digits 1, 2, 3, 4, 5, 6 and 7 such that no digit appears more than once, and '123' always appear as a string?

Correct option is A

Given:

We are given the digits:
{1, 2, 3, 4, 5, 6, 7}

We are to form 4-digit numbers such that:

• No digit is repeated
• The substring ‘123’ must appear together and in the same order

Solution:

1. Treat '123' as a single block
• Since '123' must appear as a block, consider it as one unit of 3 digits.
2. Choose one more digit to make a 4-digit number
• Remaining digits available for selection: {4, 5, 6, 7}
• Cannot use 1, 2, or 3 again (due to no repetition rule)
• So, 4 choices available for the extra digit
3. Count valid positions to place the block ‘123’
• In a 4-digit number, a block of 3 digits can be placed in only 2 ways:
• At the beginning: Format = ‘123X’
• At the end: Format = ‘X123’
4. Total combinations:
• 2 valid positions × 4 choices (for the extra digit)
  = 8 valid 4-digit numbers

Final Answer: (a) 8