A class has 12 boys and 4 girls. If three students are selected at random from the class,
what is the probability that all will be boys?

Options

A. `1/55`
B. `1/4`
C. `1/3`
D. `11/28`

Explanation

To find the probability that all three selected students are boys from a class of 12 boys and 4 girls, we can follow these steps:

### Step 1: Determine the total number of students
The total number of students in the class is:
\[ 12 \text{ (boys)} + 4 \text{ (girls)} = 16 \text{ (total students)} \]

### Step 2: Calculate the probability of selecting a boy on the first selection
The probability of selecting a boy first is:
\[ P(\text{1st boy}) = \frac{\text{Number of boys}}{\text{Total number of students}} = \frac{12}{16} \]

### Step 3: Update the number of boys and total students after the first selection
After selecting one boy, the remaining students are:
- Boys: \(12 - 1 = 11\)
- Girls: \(4\)
- Total remaining students: \(16 - 1 = 15\)

### Step 4: Calculate the probability of selecting a boy on the second selection
The probability of selecting a boy second is:
\[ P(\text{2nd boy}) = \frac{11}{15} \]

### Step 5: Update the number of boys and total students after the second selection
After selecting another boy, the remaining students are:
- Boys: \(11 - 1 = 10\)
- Girls: \(4\)
- Total remaining students: \(15 - 1 = 14\)

### Step 6: Calculate the probability of selecting a boy on the third selection
The probability of selecting a boy third is:
\[ P(\text{3rd boy}) = \frac{10}{14} \]

### Step 7: Calculate the combined probability of all three selections being boys
The combined probability of all three selections being boys is the product of the individual probabilities:
\[ P(\text{all boys}) = P(\text{1st boy}) \times P(\text{2nd boy}) \times P(\text{3rd boy}) \]
\[ = \frac{12}{16} \times \frac{11}{15} \times \frac{10}{14} \]

### Step 8: Simplify the expression
Now, we can simplify the expression:
\[ = \frac{12 \times 11 \times 10}{16 \times 15 \times 14} \]
We can simplify this step-by-step:
- \( \frac{12}{16} = \frac{3}{4} \)
- \( \frac{10}{14} = \frac{5}{7} \)

So now we have:
\[ = \frac{3}{4} \times \frac{11}{15} \times \frac{5}{7} \]

Now multiplying:
\[ = \frac{3 \times 11 \times 5}{4 \times 15 \times 7} \]
\[ = \frac{165}{420} \]

### Step 9: Further simplify
Now, we can simplify \( \frac{165}{420} \):
Both numbers can be divided by 15:
\[ = \frac{11}{28} \]

### Final Answer
Thus, the probability that all three selected students are boys is:
\[ \frac{11}{28} \]