Answer:
There are infinitely many solutions. Start with five data, in size order: a < b < c < d < e. Then c=11, the median. To get the mode of 12, we must have d=e=12. Thus far, we have (a+b+11+24)/5 = 10 the mean. But now we have run out of requirements, and we still have two unknowns. They satisfy a+b+35 = 50. Thus, pick any two values a not equal to b that add to 15 and are both less than 11.
The full set of solutions are points (a,b) run along a line joining the points (4,11) and (11,4) on graph paper, except (7.5, 7.5) and the endpoints. That would make competing modes. Try a=5.6 and b = 9.4 for instance!
