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)
- 4897
- 4898
- 4896
- 4899
The Correct Option is B
Approach Solution - 1
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 \( 4898 \).
Approach Solution -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
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.