Question

Using the number 1, 2, 3 ... 7, total numbers of 7 digit number which does not contain string 154 or 2367 is (Repetition is not allowed)

• A. 4897
• B. 4898
• C. 4896
• D. 4899

Answer 1

The Correct Option is B

The numbers are \( 1, 2, 3, 4, 5, 6, 7 \).

• Step 1: Numbers having the string \( (154) \):
  • Positions: \( (154), 2, 3, 6, 7 \)
  • Total permutations: \( 5! = 120 \)
• Step 2: Numbers having the string \( (2467) \):
  • Positions: \( (2467), 1, 3, 5 \)
  • Total permutations: \( 4! = 24 \)
• Step 3: Numbers having both strings \( (154) \) and \( (2467) \):
  • Positions: \( (154), (2467) \)
  • Total permutations: \( 2! = 2 \)
• Step 4: Apply the Inclusion-Exclusion Principle:

\( n((154) \cup (2467)) = 5! + 4! - 2! \).

• Simplify:

\( n((154) \cup (2467)) = 120 + 24 - 2 = 142 \).

• Step 5: Total numbers:

Total permutations: \( 7! = 5040 \).

• Step 6: Numbers having neither \( (154) \) nor \( (2467) \):

Required numbers = \( 5040 - 142 = 4898 \).

Final Answer: The required numbers are \( 4898 \).

Answer 2

The correct option is (B): 4898

Total numbers - when 154 comes as a n string-when 2367 comes as+2 a string

7!-5!-4!+2

5040-120-24+2

=4898