Do they have to be different? In other words, will "10+10+10+10+10+10" do as an option?
The full mathematical analysis of this is complicated and requires the use of generating functions. An interesting treatise on this subject is given in Polya & Szego's "Problems and Theorems in Analysis" (Springer-Verlag 1924, reprinted 1970).
However, it's my guess that this question is being asked from a somewhat more basic level than postgrad (we never did generating functions even in my MMath course) so I presume the question you're being asked to solve requires a more basic technique, such as hunting for the solutions by exhaustion.
So, I'd start by going:
1+2+3+4+5+(whatever 60-1-2-3-4-5 is)
1+2+3+4+6+(whatever 60-1-2-3-4-6 is, I'll leave you to do the tricky technical higher mathematics here)
1+2+3+4+7+ ... etc.
Long job, but let's face it, it's Saturday night and you're doing maths so it's not as if you've got a date or anything.
Edit: Sorry, I am an idiot, please disregard this
It seems to me that this might be a very long job, as I think there are infinitely many. If we call the sum of the first 5 integers x then a 6th integer 60-x will always exist to make 60. The restriction that integers can't be repeated doesn't seem severe enough to prevent this being the case, although I haven't proved this.Long job, but let's face it, it's Saturday night and you're doing maths so it's not as if you've got a date or anything.