Sure, let's solve the given problem step by step:
### 1. Finding the value of [tex]\(6!\)[/tex]:
To find the number of ways six people can be placed in a line for a photo, we use the factorial notation [tex]\(6!\)[/tex], which means multiplying all whole numbers from 1 to 6. Therefore, the calculation is as follows:
[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \][/tex]
So, the value of [tex]\(6!\)[/tex] is:
[tex]\[ \boxed{720} \][/tex]
### 2. Finding the number of ways to choose two people for specific roles:
When choosing two out of six people to perform specific roles (where order matters), we use permutations. The expression provided is:
[tex]\[ \frac{6!}{(6-2)!} \][/tex]
Let's simplify this:
[tex]\[ \frac{6!}{4!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} = 6 \times 5 = 30 \][/tex]
So, the number of ways to choose the two people is:
[tex]\[ \boxed{30} \][/tex]
### 3. Finding the number of ways to choose a group of three people:
To determine the number of ways to choose a group of three people from six (where the order does not matter), we use combinations. The expression given is:
[tex]\[ \binom{6}{3} = \frac{6!}{(6-3)! \times 3!} \][/tex]
Let's simplify this:
[tex]\[ \binom{6}{3} = \frac{6!}{3! \times 3!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 3 \times 2 \times 1} = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \][/tex]
So, the number of ways to choose the group of three people is:
[tex]\[ \boxed{20} \][/tex]
