If the five numbers in ascending order are a, b, c, d, e, then c=15 because the middle value is the median. The mode is 17, so d=e=17 because d and e have to be greater than the median.
The mean is 10 so the total sum is 50. Therefore a+b=50-(15+17+17)=1.
The range is e-a=17-a=30, so a=-13, making b=1-a=14.
The 5 numbers are -13, 14, 15, 17, 17.