From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $$X$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$n - m$$ is equal to ___________

We need the variance of $$X$$ (number of defectives in a sample of 5 from 12 items with 3 defectives, drawn without replacement). Here, $$X$$ follows a Hypergeometric distribution with parameters $$N=12$$, $$K=3$$ (defectives), and $$n=5$$ (sample size). The variance for a Hypergeometric distribution is given by $$\text{Var}(X) = n \cdot \frac{K}{N} \cdot \frac{N-K}{N} \cdot \frac{N-n}{N-1}$$.

Substituting the values yields $$= 5 \cdot \frac{3}{12} \cdot \frac{9}{12} \cdot \frac{7}{11} = \frac{5 \times 3 \times 9 \times 7}{12 \times 12 \times 11} = \frac{945}{1584}.$$ Since $$\gcd(945, 1584) = 9$$, we have $$\frac{945}{1584} = \frac{105}{176}.$$ Verifying the factors, $$105 = 3\times5\times7$$ and $$176 = 2^4\times11$$, confirms that $$\gcd(105,176)=1$$.

Setting $$m=105$$ and $$n=176$$ gives $$n - m = 176 - 105 = 71$$.

The correct answer is 71.