we have no's from 1 to 50. So the minimum sum that can be obtained by picking 10 numbers is 1+2+3...10= 55
maximum sum that can be obtained is 41+42+43+.....50=455
so our range of sum is from 55 to 455
now lets assume that no of ways to pick 10 numbers from these 50 numbers is S
We have to calculate the probabilities of the respective sums
so for 55 there is only 1 way of picking 10 numbers such that their sum is 55
similarly there is only 1 way of picking 10 numbers such that the sum is 455
now we have to calculate 
where x is the sum and f(x) is the respective probability
now the denominator of this summation will S(as assumed earlier). we only need to worry about the numerator
in the numerator the terms would be like 1*55 + 1*56 +.............+1*454 +1*455. now this sequence is symmetric .So lets pair up the numbers like(1st and last, 2nd and 2nd last ...) so we will get something like 1*510 +1*510+........ (NOTE : we will have terms like 2*510 ,3*510 etc also)
now if we observe closely this would sum up to 510*(S/2).
so our final answer would be (510*(S/2))/S= 510/2 =255
If this is still not clear how S/2 comes take up smaller examples and see for urself
