How many 5-digit prime numbers can be formed using the digits...

Any 5-digits number formed by the digits 1,2,3,4 and 5 (without repetition) will be always divisible by number 3 , because sum of digits of any number, thus formed $1+2+3+4+5=15$, is divisible by 3. Hence, prime number of 5 digits can not be obtained by using the digits $1,2,3,4,5$.