1. Identify the distribution

Since 5 items are drawn without replacement from 12 items containing 3 defectives, the number of defectives in the sample, $X$, follows a hypergeometric distribution with:

• Population size: $N=12$
• Number of defectives in population: $K=3$
• Sample size: $r=5$

So, $X \sim \text{Hypergeometric}(N=12,K=3,r=5).$

2. Use the variance formula for hypergeometric distribution

For a hypergeometric random variable, $\operatorname{Var}(X)=r\cdot \frac{K}{N}\left(1-\frac{K}{N}\right)\cdot \frac{N-r}{N-1}.$

Substitute the values: $\operatorname{Var}(X)=5\cdot \frac{3}{12}\left(1-\frac{3}{12}\right)\cdot \frac{12-5}{12-1}.$

3. Simplify step by step

First, $\frac{3}{12}=\frac14, \qquad 1-\frac14=\frac34.$

Also, $\frac{12-5}{12-1}=\frac{7}{11}.$

Therefore, $\operatorname{Var}(X)=5\cdot \frac14\cdot \frac34\cdot \frac{7}{11}.$

Now, $\frac14\cdot \frac34=\frac{3}{16}.$

Hence, \operatorname{Var}(X)=5\cdot \frac{3}{16}\cdot \frac{7}{11}=rac{105}{176}.

Thus, $\frac{m}{n}=\frac{105}{176}.$

Since $\gcd(105,176)=1$, we get $m=105,\quad n=176.$

4. Compute $n-m$

$n-m=176-105=71.$

5. Comparison with stored answer

Stored correct answer = 71, which matches our result.