To address the given problem, we need to understand combinations and how to calculate them. Let's break down the solution step-by-step.
Step 1: Calculate the total number of ways to choose 3 people out of 8
The problem involves combinations since the order in which people are chosen does not matter. The formula for combinations is given by:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
Here, [tex]\( n = 8 \)[/tex] (total people) and [tex]\( k = 3 \)[/tex] (people to be chosen).
So, the number of ways to choose 3 people out of 8 is:
[tex]\[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} \][/tex]
After calculation, we find:
[tex]\[ C(8, 3) = 56 \][/tex]
So, there are 56 different ways to choose 3 people out of 8. This matches option D:
[tex]\[ \boxed{56} \][/tex]
Step 2: Calculate the number of ways to be chosen along with your friend
Since you and your friend are already chosen, we need to choose 1 more person out of the remaining 6 people (since 2 people are already chosen out of the 8).
Thus, the number of ways to choose 1 person from 6 people is:
[tex]\[ C(6, 1) = \frac{6!}{1!(6-1)!} = \frac{6!}{1!5!} \][/tex]
After calculation, we find:
[tex]\[ C(6, 1) = 6 \][/tex]
So, there are 6 ways in which you and your friend can both be chosen, along with one other person from the remaining 6.
As a final step, consider the given choices to identify the correct options:
- Option A: This reflects a permutation problem, not suitable here.
- Option B: This is an invalid combination notation.
- Option C: This is a permutation problem addressing a different concept.
- Option D: This is the correct total combination calculation.
Thus, the total number of ways to choose 3 people out of 8 is correctly given by 56 (option D), and the number of ways you and your friend can both be among those chosen is 6.
