A five-digit number is formed by the digit 1, 2, 3, 4, 5 without repetition. Find the probability that the number formed is divisible by 4.

Explanation

To solve the problem of finding the probability that a five-digit number formed by the digits 1, 2, 3, 4, and 5 (without repetition) is divisible by 4, we can follow these steps:

### Step 1: Determine the total number of five-digit numbers that can be formed.
Since we are using the digits 1, 2, 3, 4, and 5 without repetition, the total number of five-digit numbers can be calculated using the factorial of the number of digits.

\[ \text{Total numbers} = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 2: Identify the condition for divisibility by 4.
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Therefore, we need to find all pairs of the last two digits from the digits 1, 2, 3, 4, and 5 that are divisible by 4.

### Step 3: List the possible pairs of last two digits.
We will check all combinations of the last two digits formed by the digits 1, 2, 3, 4, and 5:

- 12 (not divisible by 4)
- 13 (not divisible by 4)
- 14 (not divisible by 4)
- 15 (not divisible by 4)
- 21 (not divisible by 4)
- 23 (not divisible by 4)
- 24 (divisible by 4)
- 25 (not divisible by 4)
- 31 (not divisible by 4)
- 32 (divisible by 4)
- 34 (not divisible by 4)
- 35 (not divisible by 4)
- 41 (not divisible by 4)
- 42 (not divisible by 4)
- 43 (not divisible by 4)
- 45 (not divisible by 4)
- 51 (not divisible by 4)
- 52 (divisible by 4)
- 53 (not divisible by 4)
- 54 (not divisible by 4)

The valid pairs of last two digits that are divisible by 4 are: **24, 32, and 52**.

### Step 4: Count the favorable outcomes.
For each valid pair of last two digits, we will calculate how many five-digit numbers can be formed:

1. **Last two digits = 24**: Remaining digits are 1, 3, 5.
- The number of arrangements = 3! = 6.

2. **Last two digits = 32**: Remaining digits are 1, 4, 5.
- The number of arrangements = 3! = 6.

3. **Last two digits = 52**: Remaining digits are 1, 3, 4.
- The number of arrangements = 3! = 6.

Total favorable outcomes = \(6 + 6 + 6 = 18\).

### Step 5: Calculate the probability.
The probability \(P\) that a randomly formed five-digit number is divisible by 4 is given by the ratio of the number of favorable outcomes to the total number of outcomes.

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{18}{120} = \frac{3}{20} \]

Thus, the required probability is:

\[ \boxed{\frac{3}{20}} \]