Certainly! Let's work through the problem step by step.
First, we need to determine the total number of students who attended the assembly. We do this by summing the number of students from each class:
- Freshmen: 31
- Sophomores: 10
- Juniors: 17
- Seniors: 22
Adding these together:
[tex]\[ 31 + 10 + 17 + 22 = 80 \][/tex]
So, there are a total of 80 students.
Next, we calculate the probability that the first student chosen is a senior. There are 22 seniors out of the 80 students. Thus, the probability [tex]\( P(\text{senior}) \)[/tex] is:
[tex]\[ P(\text{senior}) = \frac{22}{80} = 0.275 \][/tex]
The first student then returns to the group, so the total number of students remains 80. Now, calculate the probability that the second student chosen is a sophomore. There are 10 sophomores out of the 80 students. Thus, the probability [tex]\( P(\text{sophomore}) \)[/tex] is:
[tex]\[ P(\text{sophomore}) = \frac{10}{80} = 0.125 \][/tex]
Finally, to find the combined probability that the first student chosen is a senior and the second student chosen is a sophomore, we multiply these two probabilities:
[tex]\[ P(\text{senior and sophomore}) = P(\text{senior}) \times P(\text{sophomore}) = 0.275 \times 0.125 = 0.034375 \][/tex]
To express this probability as a fraction, we see that:
[tex]\[ 0.034375 = \frac{11}{320} \][/tex]
Thus, the probability that the first student chosen is a senior and the second student chosen is a sophomore is:
[tex]\[ \frac{11}{320} \][/tex]
Therefore, the correct answer is:
[tex]\(\frac{11}{320}\)[/tex].
