What are the chances that 4 even numbers are picked, followed by 1 odd number, and that none of them chose the same number.
For the first number picked there is a 5/10 chance that it will be even.
For the second number picked there is a 4/10 chance that it will be even AND different to the previous numbers.
For the third number there is a 3/10 chance that it will be even and different.
For the fourth number there is a 2/10 chance that the number will be even and different, and finally
for the fifth number there is a 5/10 chance that it will be odd (and if so is clearly different to the previous numbers).
The chance of all these being true is:
5/10 * 4/10 * 3/10 * 2/10 * 5/10. = 3/50