I have a set S of size n whose members each have associated with them a number in the range 0.00 to 1.00 inclusive.

I want to select a subset T of size m with this property:

• the average of the numbers associated with the members of T must fall within a specified range x to y (for example, 0.65 to 0.75). Expressed differently: the sum of the numbers associated with the members of T must fall within a specified range (for example, 0.65m to 0.75m)

Further, out of all the possible Ts for a given S, x, y, I want to choose one (uniformly) randomly.

My current method is to randomly select m members of S, and then check if the sum falls in the desired range. I repeat until I get a satisfactory result. Is there an algorithm (possibly dynamic programming?) to get the desired subset without using guess and check?

Example:

S is a set of n = 200 questions, each assigned a difficulty rating between 0 and 1 inclusive. I want to generate a test T with m = 50 questions where the average difficulty is between 0.65 and 0.75 inclusive. Furthermore I want to select a (uniformly) random T, out of all the possible T‘s that satisfy my conditions.

Another Example:

S = {0.1, 0.3, 0.5, 0.8, 1.0}
n = 5
m = 3
x = 0.5
y = 0.75

All possible T and their average value
{0.1, 0.3, 0.5} = 0.8 / 3 = 0.26
{0.1, 0.3, 0.8} = 1.2 / 3 = 0.40
{0.1, 0.3, 1.0} = 1.4 / 3 = 0.46
{0.1, 0.5, 0.8} = 1.4 / 3 = 0.46
{0.1, 0.5, 1.0} = 1.6 / 3 = 0.53
{0.1, 0.8, 1.0} = 1.9 / 3 = 0.63
{0.3, 0.5, 0.8} = 1.6 / 3 = 0.53
{0.3, 0.5, 1.0} = 1.8 / 3 = 0.60
{0.3, 0.8, 1.0} = 2.1 / 3 = 0.70
{0.5, 0.8, 1.0} = 2.3 / 3 = 0.76

Subsets of S with size m with average values between x and y
{0.1, 0.5, 1.0}
{0.1, 0.8, 1.0}
{0.3, 0.5, 0.8}
{0.3, 0.5, 1.0}
{0.3, 0.8, 1.0}

I am trying to come up with an algorithm to produce one of these 5 subsets at random, without first calculating every subset of S with size m. It seems that guess and check is the best method.