A number k is selected from the set {1, 2, 3, 4, ……10}. If `k^(2)-4k+3lt0` then the probability is

Options

A. `1/5`
B. `2/5`
C. `3/10`
D. `1/10`

Answer

To solve the problem, we need to determine the values of \( k \) from the set \( \{1, 2, 3, 4, \ldots, 10\} \) that satisfy the inequality \( k^2 - 4k + 3 < 0 \).

### Step-by-Step Solution:

1. **Identify the quadratic expression**:
The expression we are working with is \( k^2 - 4k + 3 \).

2. **Factor the quadratic expression**:
We can factor \( k^2 - 4k + 3 \) as follows:
\[ k^2 - 4k + 3 = (k - 1)(k - 3) \]

3. **Set up the inequality**:
We need to solve the inequality:
\[ (k - 1)(k - 3) < 0 \]

4. **Find the critical points**:
The critical points of the expression are obtained by setting each factor to zero:
\[ k - 1 = 0 \quad \Rightarrow \quad k = 1 \]
\[ k - 3 = 0 \quad \Rightarrow \quad k = 3 \]

5. **Test intervals**:
We will test the intervals determined by the critical points \( k = 1 \) and \( k = 3 \):
- For \( k < 1 \) (e.g., \( k = 0 \)): \( (0 - 1)(0 - 3) = 3 > 0 \)
- For \( 1 < k < 3 \) (e.g., \( k = 2 \)): \( (2 - 1)(2 - 3) = 1 \cdot (-1) = -1 < 0 \)
- For \( k > 3 \) (e.g., \( k = 4 \)): \( (4 - 1)(4 - 3) = 3 \cdot 1 = 3 > 0 \)

6. **Determine the solution set**:
The inequality \( (k - 1)(k - 3) < 0 \) holds true for the interval \( 1 < k < 3 \). Therefore, the integer values of \( k \) that satisfy this inequality are:
\[ k = 2 \]

7. **Count the total outcomes**:
The total number of elements in the set \( \{1, 2, 3, \ldots, 10\} \) is 10.

8. **Count the favorable outcomes**:
The only favorable outcome is \( k = 2 \), which is 1 favorable outcome.

9. **Calculate the probability**:
The probability \( P \) that a randomly selected \( k \) satisfies the inequality is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{10} \]

### Final Answer:
The probability that \( k \) satisfies the inequality \( k^2 - 4k + 3 < 0 \) is \( \frac{1}{10} \).