There's two ways of tackling this that i know of:

**Use the multinomial distribution**

The problem you are describing can be described in terms of a standard distribution called the multinomial distribution, which you can look up. If your not comfortable with math you may find the notation difficult to follow though.

**Use a generating function**

this is easyish to do but hard to understand.

The number of permutations scoring in a test with questions is the coefficient of in the expanded version of:

You can then ratio these to get probabilities and hence work out the chance of falling in any particular range.

__Worked Example__

Lets take the case of N=3 as an example, our expansion is:

eg, the number of permuations resulting a score of 16 is the coefficient of

which in the above expansion has been calculated as 3.

The total number of permutations will be the sum of the coefficients: 1+3+3+4+6+3+3+3+1=27

So, the probability of scoring 16 would be 3/27

In this case:

P(Score 12)=1/27

P(Score 16)=3/27

P(Score 20)=3/27

P(Score 24)=4/27

P(Score 28)=6/27

...etc

Now to find the score that 33% fall short of. (33% is the threshold probability you posted for SR1). Just add up probabilities of the lowest score until you get past 33%. The cumulative probabilities are:

P(Score 12 or less) = 1/27

P(Score 16 or less) = 1/27 + 3/27 = 14.8%

P(Score 20 or less) = 1/27 + 3/27 + 3/27 = 25.9%

P(Score 24 or less) = 1/27 + 3/27 + 3/27 +4/27 = 40.7%

P(Score 28 or less) = 1/27 + 3/27 + 3/27 +4/27 + 6/27 = 63%

...

So, your SR1 limit would be a score of either 20 or 24 (depending on whether you want the probability to be too low or too high).

Also, since your post sounds like a real world situation involving test results...you should not rely on anything i say without verifying it first.