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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 (1998 - 2 Marks)
- a)6
- b)7
- c)8
- d)9
Correct answer is option 'B'. Can you explain this answer?
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An n-digit number is a positive number with exactly n digits.Nine hund...
(b) Given digits are 2,5 and 7 and n-digit number is to be formed using these digits. Out of n places, each place can be filled in 3 ways. Thus total number of ways = 3^n . Now, 3^n≥ 900 = 3^2*100 ⇒ n – 2 ≥ 5 .
Least value of n is 7.
Most Upvoted Answer
An n-digit number is a positive number with exactly n digits.Nine hund...
Problem Analysis:
We need to find the smallest value of n for which we can form 900 distinct n-digit numbers using only the digits 2, 5, and 7.
Key Observations:
1. Since we have three digits available (2, 5, and 7), the total number of distinct n-digit numbers that can be formed is 3^n.
2. For any number of digits n, the maximum number of distinct n-digit numbers that can be formed using the digits 2, 5, and 7 is 3^n.
3. To find the smallest value of n for which we can form 900 distinct n-digit numbers, we need to find the smallest n such that 3^n >= 900.
Solution:
To find the smallest n such that 3^n >= 900, we can use trial and error or logarithms.
Using Trial and Error:
Let's start with n = 1 and calculate 3^n for each value of n until we find a value that is greater than or equal to 900.
n = 1: 3^1 = 3 (less than 900)
n = 2: 3^2 = 9 (less than 900)
n = 3: 3^3 = 27 (less than 900)
n = 4: 3^4 = 81 (less than 900)
n = 5: 3^5 = 243 (less than 900)
n = 6: 3^6 = 729 (less than 900)
n = 7: 3^7 = 2187 (greater than 900)
Therefore, the smallest value of n for which 3^n is greater than or equal to 900 is n = 7.
Using Logarithms:
We can also solve this problem using logarithms.
We need to find the smallest n such that 3^n >= 900.
Taking the logarithm of both sides, we get:
n >= log(base 3) 900
Using a calculator or logarithm tables, we find that log(base 3) 900 is approximately 6.431.
Since n must be an integer, the smallest integer value greater than or equal to 6.431 is 7.
Therefore, the smallest value of n for which 3^n is greater than or equal to 900 is n = 7.
Conclusion:
The smallest value of n for which we can form 900 distinct n-digit numbers using only the digits 2, 5, and 7 is n = 7.
We need to find the smallest value of n for which we can form 900 distinct n-digit numbers using only the digits 2, 5, and 7.
Key Observations:
1. Since we have three digits available (2, 5, and 7), the total number of distinct n-digit numbers that can be formed is 3^n.
2. For any number of digits n, the maximum number of distinct n-digit numbers that can be formed using the digits 2, 5, and 7 is 3^n.
3. To find the smallest value of n for which we can form 900 distinct n-digit numbers, we need to find the smallest n such that 3^n >= 900.
Solution:
To find the smallest n such that 3^n >= 900, we can use trial and error or logarithms.
Using Trial and Error:
Let's start with n = 1 and calculate 3^n for each value of n until we find a value that is greater than or equal to 900.
n = 1: 3^1 = 3 (less than 900)
n = 2: 3^2 = 9 (less than 900)
n = 3: 3^3 = 27 (less than 900)
n = 4: 3^4 = 81 (less than 900)
n = 5: 3^5 = 243 (less than 900)
n = 6: 3^6 = 729 (less than 900)
n = 7: 3^7 = 2187 (greater than 900)
Therefore, the smallest value of n for which 3^n is greater than or equal to 900 is n = 7.
Using Logarithms:
We can also solve this problem using logarithms.
We need to find the smallest n such that 3^n >= 900.
Taking the logarithm of both sides, we get:
n >= log(base 3) 900
Using a calculator or logarithm tables, we find that log(base 3) 900 is approximately 6.431.
Since n must be an integer, the smallest integer value greater than or equal to 6.431 is 7.
Therefore, the smallest value of n for which 3^n is greater than or equal to 900 is n = 7.
Conclusion:
The smallest value of n for which we can form 900 distinct n-digit numbers using only the digits 2, 5, and 7 is n = 7.
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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 (1998 - 2 Marks)a)6b)7c)8d)9Correct answer is option 'B'. Can you explain this answer?
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