Introduction to the Problem
" "When selecting a three-digit number at random, it is often interesting to determine the probability that the number has no repeated digits. This question can be approached through simple logical reasoning and through computational methods such as the J programming language. In this article, we will explore both methods to arrive at the same solution.
" "Brute Force Solution Using the J Programming Language
" "One effective way to solve this problem is by using a programming language like J. In J, we can write a concise expression to find such numbers. Let's dive in with the J code:
" "/3ea~.ea sep ea 100 to 999 " "648 900 " "The probability is 648/900 0.72 72" "
The output shows that out of 900 three-digit numbers, 648 have no repeated digits. Thus, the probability is simply ( frac{648}{900} 0.72 72%).
" "Logical Solution
" "Another way to solve this problem involves some basic logical reasoning. We start by considering the constraints on the digits of the number. Let's break it down step by step:
" "Step 1: First digit: It cannot be 0 (since the number is three digits long). Thus, we have 9 options for the first digit (1 through 9).
" "Step 2: Second digit: It can be any digit from 0 to 9, except the digit we used for the first position. Therefore, we have 9 options left.
" "Step 3: Third digit: It can be any digit from 0 to 9, except the two digits already used. Thus, we have 8 options left.
" "The total number of valid three-digit numbers without repeating digits is the product of the options available for each position:
" "9 (for the first digit) x 9 (for the second digit) x 8 (for the third digit) 648.
" "There are a total of 900 possible three-digit numbers (100 to 999). Therefore, the probability of selecting a three-digit number with no repeated digits is:
" "[ frac{648}{900} 0.72 72% ]
" "Mathematical Formula
" "From a mathematical perspective, we can express the total number of three-digit numbers that do not repeat digits as:
" "N 9 cdot 10 cdot 8 648
" "The total number of three-digit numbers is 900. Thus, the probability of selecting a non-repeating number is:
" "[ mathcal{P} frac{648}{900} boxed{frac{18}{25}} ]
" "Conclusion
" "In conclusion, we have explored both a brute force approach using the J programming language and a logical approach to determine the probability of selecting a three-digit number with no repeated digits. The answer to this problem is 72%, which can be expressed as a fraction (frac{18}{25}).