1. Interpret the condition

We choose $3$ numbers from $\{1,2,3,4,5,6,7,8\}$ without replacement.

We need: $P(\text{minimum }=3\mid \text{maximum }=6).$

Using conditional probability, $P(A\mid B)=\frac{\#(A\cap B)}{\#(B)},$ where:

• $A$: minimum is $3$
• $B$: maximum is $6$

2. Count all selections with maximum $6$

If the maximum is $6$, then:

• $6$ must be included,
• the other two numbers must be chosen from $\{1,2,3,4,5\}$.

So the total number of such selections is $\binom{5}{2}=10.$

3. Count selections with maximum $6$ and minimum $3$

For the minimum to be $3$:

• $3$ must be included,
• no number smaller than $3$ can be chosen,
• since the maximum is $6$, $6$ must also be included.

Thus the third number must be chosen from $\{4,5\}$.

Number of favorable selections: $\binom{2}{1}=2.$ These sets are: $\{3,4,6\},\ \{3,5,6\}.$

4. Compute the conditional probability

$P(\text{minimum }=3\mid \text{maximum }=6)=\frac{2}{10}=\frac{1}{5}.$

5. Match with options

$\frac{1}{5}$ corresponds to Option B.

6. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So the derived answer agrees with the stored answer.