From the set `{1,2,3,4,5}` two numbers a and b `(a != b)` are chosen at random . The probability that `a/b` is an integar is

Options

A. `1/3`
B. `1/4`
C. `1/2`
D. `3/5`

Answer

To solve the problem, we need to find the probability that the ratio \( \frac{a}{b} \) is an integer when two different numbers \( a \) and \( b \) are chosen from the set \{1, 2, 3, 4, 5\}.

### Step-by-Step Solution:

1. **Identify the Sample Space**:
We need to find all possible pairs \( (a, b) \) where \( a \neq b \). The set has 5 elements, so the total number of ways to choose 2 different numbers from the set is given by:
\[ \text{Total pairs} = 5 \times 4 = 20 \]
(since for each choice of \( a \), there are 4 choices for \( b \)).

2. **List the Favorable Outcomes**:
We need to find pairs \( (a, b) \) such that \( \frac{a}{b} \) is an integer. This occurs when \( a \) is a multiple of \( b \). We will check each possible value of \( b \) and find corresponding values of \( a \):
- If \( b = 1 \): \( a \) can be \( 2, 3, 4, 5 \) (4 outcomes)
- If \( b = 2 \): \( a \) can be \( 2, 4 \) (1 outcome)
- If \( b = 3 \): \( a \) can be \( 3 \) (1 outcome)
- If \( b = 4 \): \( a \) can be \( 4 \) (1 outcome)
- If \( b = 5 \): \( a \) can be \( 5 \) (1 outcome)

Now, we can summarize the favorable outcomes:
- \( (2, 1) \)
- \( (3, 1) \)
- \( (4, 1) \)
- \( (5, 1) \)
- \( (4, 2) \)

Counting these, we find there are 5 favorable outcomes.

3. **Calculate the Probability**:
The probability \( P \) that \( \frac{a}{b} \) is an integer is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{20} = \frac{1}{4} \]

### Final Answer:
Thus, the probability that \( \frac{a}{b} \) is an integer is \( \frac{1}{4} \).