Certainly! Let's work through the problem step-by-step to find the probability that the coach chooses all girls for the competition.
### Step 1: Identify the total number of students and the number of girls
- Total students: There are 4 boys and 8 girls on the debate team, so the total number of students is [tex]\(4 + 8 = 12\)[/tex].
- Girls: There are 8 girls.
### Step 2: Determine the number of students chosen
- The coach randomly chooses 3 students.
### Step 3: Calculate the number of ways to choose 3 girls from 8
We use combinations to find this. The number of ways to choose 3 girls out of 8 is given by the combination formula [tex]\( \binom{n}{k} \)[/tex], where [tex]\( n \)[/tex] is the total number and [tex]\( k \)[/tex] is the number chosen.
[tex]\[ \binom{8}{3} \][/tex]
### Step 4: Calculate the total number of ways to choose 3 students from 12
Similarly, we use the combination formula to find the number of ways to choose 3 students from the total 12 students:
[tex]\[ \binom{12}{3} \][/tex]
### Step 5: Write down the numbers of combinations obtained
From our calculations:
- The number of ways to choose 3 girls from 8 is [tex]\( 56 \)[/tex].
- The total number of ways to choose 3 students from 12 is [tex]\( 220 \)[/tex].
### Step 6: Calculate the probability
The probability that the coach chooses all girls is the number of favorable outcomes (ways to choose 3 girls) divided by the total number of possible outcomes (ways to choose any 3 students).
[tex]\[ \text{Probability} = \frac{\text{Number of ways to choose 3 girls}}{\text{Total number of ways to choose 3 students}} = \frac{56}{220} \][/tex]
### Step 7: Simplify the fraction
Now, we simplify the fraction:
[tex]\[ \frac{56}{220} = \frac{28}{110} = \frac{14}{55} \][/tex]
Therefore, the probability that the coach chooses all girls is [tex]\( \frac{14}{55} \)[/tex].
So, the final answer is:
[tex]\[ \boxed{\frac{14}{55}} \][/tex]
