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Chapter 5: Problem 430

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

Expert verified
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

01

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 \(3^n\).
02

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 \(3^n \geq 900\).
03

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: \[n \geq \log_3{(900)}\]
04

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.
05

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.

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