A three digit number is written down by random choice of the digits 1 to 9 with replacements. The probability that atleast one of the digits chosen is a perfect square is

Options

A. `8/27`
B. `4/9`
C. `2/9`
D. None of these

Answer

To solve the problem of finding the probability that at least one of the digits chosen in a three-digit number (formed by random selection of digits from 1 to 9 with replacement) is a perfect square, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Perfect Square Digits**:
The digits from 1 to 9 that are perfect squares are 1, 4, and 9. Therefore, there are 3 perfect square digits.

**Hint**: List out the digits from 1 to 9 and identify which ones are perfect squares.

2. **Calculate Total Possible Outcomes**:
Since we are forming a three-digit number and each digit can be any of the 9 digits (1 to 9), the total number of possible three-digit combinations is:
\[ 9 \times 9 \times 9 = 729 \]

**Hint**: Remember that since digits can repeat, you multiply the number of choices for each digit.

3. **Calculate the Probability of Not Selecting a Perfect Square**:
The digits that are not perfect squares are 2, 3, 5, 6, 7, and 8. This gives us 6 non-perfect square digits.
The number of ways to select three digits that are all non-perfect squares is:
\[ 6 \times 6 \times 6 = 216 \]

**Hint**: Count the non-perfect square digits and apply the same multiplication principle.

4. **Calculate the Probability of Selecting At Least One Perfect Square**:
To find the probability of selecting at least one perfect square digit, we can use the complement rule:
\[ P(\text{at least one perfect square}) = 1 - P(\text{no perfect squares}) \]
The probability of selecting no perfect squares is:
\[ P(\text{no perfect squares}) = \frac{\text{Number of ways to select no perfect squares}}{\text{Total possible outcomes}} = \frac{216}{729} \]
Therefore,
\[ P(\text{at least one perfect square}) = 1 - \frac{216}{729} = \frac{729 - 216}{729} = \frac{513}{729} \]

**Hint**: Use the complement rule to simplify your calculations.

5. **Simplify the Probability**:
Now, simplify \(\frac{513}{729}\). The greatest common divisor (GCD) of 513 and 729 is 9. Thus, we can simplify:
\[ \frac{513 \div 9}{729 \div 9} = \frac{57}{81} \]
Further simplifying gives:
\[ \frac{19}{27} \]

**Hint**: Always check if the fraction can be simplified further.

### Final Answer:
The probability that at least one of the digits chosen is a perfect square is \(\frac{19}{27}\).