How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?

To solve this elegantly, I look at the available digits: 1, 2, 3, 4, and 5. The question asks for numbers formed using *all* these digits without repetition. This means I don't need to write down the numbers; I just need to understand their properties.\n\nFirst, I calculate the sum of these digits: \n1 + 2 + 3 + 4 + 5 = 15.\n\nFrom my analysis of 2019 PYQs, I know the **Divisibility Rule of 3** is a frequent favorite [7, 61]. This rule states: if the sum of digits is divisible by 3, the number itself is divisible by 3.\n\nSince the sum is **15** (which is divisible by 3), *every single number* formed by rearranging these digits will be divisible by 3. \n\nBy definition, a **Prime Number** cannot be divisible by any number other than 1 and itself [16]. Since all these 5-digit numbers are divisible by 3 (and are clearly larger than 3), they are all composite numbers, not primes.\n\nTherefore, it is impossible to form any prime number with these digits.\n\nThe count is zero.