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Directions: The answer to the following question is a single digit integer ranging from 0 to 9.
An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is
Correct answer is '7'. Can you explain this answer?
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Directions: The answer to the following question is a single digit in...
Number of distinct n-digit numbers which can be formed using digits 2, 5 and 7 = 3nWe have to find n, so that 3n ≥ 900.
⇒ 3n–2 ≥ 100
⇒ n – 2 ≥ 5
⇒ n ≥ 7
So, the least value of n is 7.
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Directions: The answer to the following question is a single digit in...
Understanding n-digit Numbers
To determine the smallest value of n for which we can form 900 distinct n-digit numbers using the digits 2, 5, and 7, we first need to analyze how many n-digit numbers can be created with these digits.
Calculating Possible Combinations
- An n-digit number can be formed where each digit can be one of three choices: 2, 5, or 7.
- The total number of n-digit combinations can be expressed as:
- 3^n
Setting Up the Equation
- We want to find the smallest n such that:
- 3^n ≥ 900
Finding the Minimum n
- Let's calculate for different values of n:
- For n = 1: 3^1 = 3
- For n = 2: 3^2 = 9
- For n = 3: 3^3 = 27
- For n = 4: 3^4 = 81
- For n = 5: 3^5 = 243
- For n = 6: 3^6 = 729
- For n = 7: 3^7 = 2187
- At n = 6, we have 729 combinations, which is insufficient. At n = 7, we reach 2187 combinations, exceeding 900.
Conclusion
- Therefore, the smallest value of n that allows for at least 900 distinct n-digit numbers using the digits 2, 5, and 7 is:
- 7
To determine the smallest value of n for which we can form 900 distinct n-digit numbers using the digits 2, 5, and 7, we first need to analyze how many n-digit numbers can be created with these digits.
Calculating Possible Combinations
- An n-digit number can be formed where each digit can be one of three choices: 2, 5, or 7.
- The total number of n-digit combinations can be expressed as:
- 3^n
Setting Up the Equation
- We want to find the smallest n such that:
- 3^n ≥ 900
Finding the Minimum n
- Let's calculate for different values of n:
- For n = 1: 3^1 = 3
- For n = 2: 3^2 = 9
- For n = 3: 3^3 = 27
- For n = 4: 3^4 = 81
- For n = 5: 3^5 = 243
- For n = 6: 3^6 = 729
- For n = 7: 3^7 = 2187
- At n = 6, we have 729 combinations, which is insufficient. At n = 7, we reach 2187 combinations, exceeding 900.
Conclusion
- Therefore, the smallest value of n that allows for at least 900 distinct n-digit numbers using the digits 2, 5, and 7 is:
- 7
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Directions: The answer to the following question is a single digit integer ranging from 0 to 9.An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible isCorrect answer is '7'. Can you explain this answer?
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