Form two digit numbers using the digits `0,1,2,3,4` without repeating the digits. Find the probability that the number so formed is greater than `30`.

Answer

To solve the problem of finding the probability that a two-digit number formed using the digits 0, 1, 2, 3, and 4 (without repeating any digits) is greater than 30, we can follow these steps:

### Step 1: Identify the available digits
The available digits are 0, 1, 2, 3, and 4.

### Step 2: Determine the total number of two-digit numbers
A two-digit number cannot start with 0. Therefore, the first digit can be one of the following: 1, 2, 3, or 4. This gives us 4 options for the first digit.

After choosing the first digit, we can choose any of the remaining digits (including 0) for the second digit. Since we cannot repeat digits, we will have 4 remaining choices for the second digit.

Thus, the total number of two-digit numbers (N(S)) that can be formed is:
\[ N(S) = \text{(choices for first digit)} \times \text{(choices for second digit)} = 4 \times 4 = 16 \]

### Step 3: List all the two-digit numbers formed
The possible two-digit numbers are:
- Starting with 1: 10, 12, 13, 14
- Starting with 2: 20, 21, 23, 24
- Starting with 3: 30, 31, 32, 34
- Starting with 4: 40, 41, 42, 43

So, the complete list of two-digit numbers is:
- 10, 12, 13, 14, 20, 21, 23, 24, 30, 31, 32, 34, 40, 41, 42, 43

### Step 4: Determine which numbers are greater than 30
Now we will identify which of these numbers are greater than 30:
- 31, 32, 34, 40, 41, 42, 43

Counting these, we find that there are 7 numbers greater than 30.

### Step 5: Calculate the probability
The probability (P) that a randomly formed two-digit number is greater than 30 is given by the ratio of the number of favorable outcomes (numbers greater than 30) to the total outcomes (total two-digit numbers formed):
\[ P(A) = \frac{N(A)}{N(S)} = \frac{7}{16} \]

### Final Answer
The probability that the number formed is greater than 30 is:
\[ \frac{7}{16} \]

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