To determine the probability that a four-person committee chosen from a group of eight boys and six girls consists of all boys, we need to follow these steps:
### Step 1: Determine the total number of students
There are 8 boys and 6 girls. Thus, the total number of students is:
[tex]\[ 8 + 6 = 14 \][/tex]
### Step 2: Calculate the total number of ways to choose 4 students from 14
We use the combination formula to calculate this:
[tex]\[
\binom{14}{4} = \frac{14!}{4!(14-4)!}
\][/tex]
Given the result from the solution, we know that:
[tex]\[
\binom{14}{4} = 1001
\][/tex]
### Step 3: Calculate the number of ways to choose 4 boys out of 8
Again, we use the combination formula:
[tex]\[
\binom{8}{4} = \frac{8!}{4!(8-4)!}
\][/tex]
Given the result from the solution, we know that:
[tex]\[
\binom{8}{4} = 70
\][/tex]
### Step 4: Calculate the probability that the committee consists of all boys
The probability is the number of favorable outcomes (choosing 4 boys out of 8) divided by the total number of outcomes (choosing any 4 students out of 14):
[tex]\[
\text{Probability} = \frac{\text{Number of ways to choose 4 boys}}{\text{Total number of ways to choose 4 students}} = \frac{70}{1001}
\][/tex]
Thus, the probability that the committee consists of all boys is:
[tex]\[
0.06993006993006994 \approx \frac{70}{1001} \approx 0.07
\][/tex]
By matching the given multiple-choice format, the fraction [tex]\(\frac{70}{1001}\)[/tex] matches with the simplified forms using the actual answer options provided in the question. It matches:
[tex]\[
\frac{10}{143}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{10}{143}}
\][/tex]
