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The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order?
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The number of ways in which eight digit number can be formed using the...
Understanding the Problem
To form an eight-digit number using the digits from 1 to 9 without repetition, we need to satisfy specific conditions: the first four digits must be in increasing order, and the last four digits must be in decreasing order.
Choosing Digits
- We select 4 digits from the 9 available digits (1 to 9).
- The remaining 5 digits will automatically be used for the last four places.
Combination Calculations
- The number of ways to select 4 digits from 9 can be calculated using combinations:
- This is represented as "9 choose 4", which equals 126.
Arranging the Digits
- Once 4 digits are chosen for the first half:
- The 4 selected digits will have only 1 way to be arranged in increasing order.
- For the last four digits:
- The remaining 5 digits can be arranged in decreasing order in only 1 way.
Final Calculation
- Thus, the total number of unique 8-digit numbers can be summarized as:
- Total ways = Number of ways to choose the digits × Ways to arrange them in specified order
- Mathematically, this can be expressed as:
- Total ways = 126 (ways to choose) × 1 (increasing arrangement) × 1 (decreasing arrangement)
Conclusion
The total number of ways to form the desired eight-digit number is 126. This method efficiently adheres to the rules of increasing and decreasing arrangements, ensuring no repetition of digits.
To form an eight-digit number using the digits from 1 to 9 without repetition, we need to satisfy specific conditions: the first four digits must be in increasing order, and the last four digits must be in decreasing order.
Choosing Digits
- We select 4 digits from the 9 available digits (1 to 9).
- The remaining 5 digits will automatically be used for the last four places.
Combination Calculations
- The number of ways to select 4 digits from 9 can be calculated using combinations:
- This is represented as "9 choose 4", which equals 126.
Arranging the Digits
- Once 4 digits are chosen for the first half:
- The 4 selected digits will have only 1 way to be arranged in increasing order.
- For the last four digits:
- The remaining 5 digits can be arranged in decreasing order in only 1 way.
Final Calculation
- Thus, the total number of unique 8-digit numbers can be summarized as:
- Total ways = Number of ways to choose the digits × Ways to arrange them in specified order
- Mathematically, this can be expressed as:
- Total ways = 126 (ways to choose) × 1 (increasing arrangement) × 1 (decreasing arrangement)
Conclusion
The total number of ways to form the desired eight-digit number is 126. This method efficiently adheres to the rules of increasing and decreasing arrangements, ensuring no repetition of digits.
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The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order?
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The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order?.
The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of ways in which eight digit number can be formed using the digits from 1 to 9 without repetition if first four places of the number are in increasing order and last four places are in decreasing order?.
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