To solve this problem, we need to calculate the number of ways to choose 3 people from a group of 7 people, with the condition that you and your friend must be among those 3 people chosen.
Let's break down the problem step by step:
1. Total People:
- There are 7 people in total, including you and your friend.
2. People to Be Chosen:
- The host needs to choose 3 people from the 7 people in the front row.
3. Condition:
- You (Person A) and your friend (Person B) must be among the 3 chosen people.
Since you and your friend must be chosen, that leaves us with 1 more spot to fill from the remaining people.
4. Remaining People:
- After selecting you and your friend, there are [tex]\(7 - 2 = 5\)[/tex] people remaining.
5. Remaining Spot:
- We need to choose 1 more person from these 5 remaining people to fill the last spot among the 3 people being chosen.
The number of ways to choose 1 person from 5 people can be represented using the combination formula:
[tex]\[
\binom{5}{1}
\][/tex]
The combination formula [tex]\(\binom{n}{k}\)[/tex] represents the number of ways to choose [tex]\(k\)[/tex] items from [tex]\(n\)[/tex] items and is defined as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n - k)!}
\][/tex]
For our case:
[tex]\[
\binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{5!}{1! \cdot 4!} = \frac{5 \cdot 4!}{1 \cdot 4!} = 5
\][/tex]
Thus, there are 5 ways to choose the 3rd person from the remaining 5 people after you and your friend have been chosen.
Therefore, the number of ways for you and your friend to both be chosen as contestants is:
[tex]\[
\boxed{5}
\][/tex]
Answer choice A ([tex]\( \binom{5}{1} = 5 \)[/tex]) is correct.
