Question

Nine hundred distinct n digit numbers are to be formed using only the 3 digits \(2,5,7 .\) The smallest value of \(\mathrm{n}\) for which this is possible is (a) 6 (b) 7 (c) 8 (d) 9

Short answer

The smallest value of n for which it is possible to form 900 distinct n-digit numbers using only the digits 2, 5, and 7 is 7. This is determined by solving the inequality \(3^n \geq 900\), which simplifies to \(n \geq \log_3{(900)}\), where n is approximately 6.29. Since n must be an integer, the smallest integer greater than or equal to 6.29 is 7. Therefore, the correct answer is (b) 7.

Step-by-step solution

Determine the number of possible combinations

The number of possible n-digit numbers using these 3 digits can be thought of as a problem of arranging the 3 digits in n distinct positions. Using the counting principle, we have 3 choices for the first position, 3 choices for the second position, and so on until the nth position. So the total number of possible combinations is \(3^n\).

Set up an inequality

We want to find the smallest n such that at least 900 n-digit numbers can be formed, so we need to find the smallest n for which \(3^n \geq 900\).

Solve the inequality

To solve the inequality \(3^n \geq 900\), we can take the logarithm base 3 of both sides: \[\log_3{(3^n)} \geq \log_3{(900)}\] By the properties of logarithms, the left side simplifies to \(n\), giving us: \[n \geq \log_3{(900)}\]

Find the smallest integer n

Calculate the approximate value for the logarithm term: \[\log_3{(900)} \approx 6.29\] Since n must be an integer, we take the smallest integer greater than or equal to 6.29, which is 7.

Choose the correct option

The smallest value of n for which it is possible to form 900 distinct n-digit numbers is 7. So the correct answer is (b) 7.

Key concepts

Counting Principle

When arranging items or forming combinations, the counting principle is a fundamental concept in combinatorics. It states that if you have a set of choices for separate actions, the total number of ways to perform all actions is the product of the number of choices for each action.

For example, if you want to form an n-digit number using the digits 2, 5, and 7, then for each digit place, you have 3 possible choices: 2, 5, or 7.

• The first digit can be any of the 3: so 3 choices.
• The second digit can also be any of the 3: again 3 choices.
• This pattern continues up to the nth digit.

Therefore, the total number of n-digit numbers you can create is given by multiplying the number of choices for each digit, leading to the expression: \(3^n\). This illustrates how the counting principle helps solve combinatorics problems efficiently.

Inequality Solving

Solving inequalities is a common mathematical task that involves finding the set of values for which the inequality holds true. In this context, we want enough numbers to fulfill a certain requirement. To reach at least 900 numbers, we set up the inequality \(3^n \geq 900\).

This approach means we're looking for the smallest number of digits, \(n\), needed to exceed or meet 900 distinct combinations using our choices. In essence, solving this inequality provides the minimum bound required.

• Identify the target: Here it's at least 900 numbers.
• Frame the inequality using available options.
• Simplify or rearrange the inequality if possible.

Solving through such logical steps ensures that none of the potential combinations are missed or underestimated.

Logarithms

Logarithms simplify the comparison of exponential growth by transforming them into linear expressions. When faced with the inequality \(3^n \geq 900\), logs help by converting the exponent into a manageable term.

Taking the log base 3 of both sides transforms the inequality into \(n \geq \log_3{(900)}\). This step is crucial because it turns the problem into a simple comparison involving \(n\). In this problem, we use the properties of logarithms such as:

• \(\log_b{b^x} = x\), which makes \(\log_3{(3^n)} = n\).
• Logs help specify the size or order of magnitude easily.

Approximating the logarithm leads to the final solution, making logs an effective tool for tackling exponential equations.

Number Formation

When forming numbers, especially in combinatorial contexts, we consider the arrangement of digits and the total possibilities. The problem's goal was to find the smallest n-digit number that could form at least 900 unique arrangements with the digits 2,5, and 7.

Here is the specific thought process behind forming numbers:

• Determine the distinct digits available, which affects uniqueness.
• Each additional digit exponentially increases the total combinations.
• Using each digit in the number contributes to forming distinct sequences.

By following these steps and understanding the principles, you can better grasp number formation and its requirements in combinatorics.