The probability that the two digit number formed by digits 1, 2, 3, 4, 5 is divisible by 4 is

Options

A. `(1)/(30)`
B. `(1)/(20)`
C. `(1)/(5)`
D. None of these

Answer

To solve the problem of finding the probability that a two-digit number formed by the digits 1, 2, 3, 4, and 5 is divisible by 4, we can follow these steps:

### Step 1: Determine the total number of two-digit numbers that can be formed.
Since we can use each of the digits (1, 2, 3, 4, 5) for both the tens and the units place, we can calculate the total number of two-digit combinations.

- **Tens place**: 5 options (1, 2, 3, 4, 5)
- **Units place**: 5 options (1, 2, 3, 4, 5)

Thus, the total number of two-digit numbers is:
\[ \text{Total two-digit numbers} = 5 \times 5 = 25 \]

### Step 2: Identify the two-digit numbers that are divisible by 4.
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. We will check each combination of digits to find those that meet this criterion.

The two-digit numbers formed by the digits 1, 2, 3, 4, and 5 are:
- 11, 12, 13, 14, 15
- 21, 22, 23, 24, 25
- 31, 32, 33, 34, 35
- 41, 42, 43, 44, 45
- 51, 52, 53, 54, 55

Now, we will check which of these numbers are divisible by 4:
- 12 (12 ÷ 4 = 3)
- 24 (24 ÷ 4 = 6)
- 32 (32 ÷ 4 = 8)
- 44 (44 ÷ 4 = 11)

Thus, the favorable cases (two-digit numbers divisible by 4) are:
- 12, 24, 32, and 44

### Step 3: Count the favorable outcomes.
From our findings, we have 4 favorable outcomes:
- 12
- 24
- 32
- 44

### Step 4: Calculate the probability.
The probability \( P \) of an event is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we found:
\[ P = \frac{4}{25} \]

### Conclusion
The probability that a two-digit number formed by the digits 1, 2, 3, 4, and 5 is divisible by 4 is \( \frac{4}{25} \).

---