Let's carefully determine the solutions to each part of the problem step by step.
### Step 1: Calculating [tex]\(6!\)[/tex]
The number of ways to place six people in a line can be determined by calculating the factorial of 6, denoted as [tex]\(6!\)[/tex]:
[tex]\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\][/tex]
Following the factorial calculation, we find:
[tex]\[6! = 720\][/tex]
Thus, the value of [tex]\(6!\)[/tex] is:
[tex]\[
\boxed{720}
\][/tex]
### Step 2: Choosing 2 people out of 6 for specific responsibilities
Two out of the six people are selected for specific responsibilities (one holding a sign and the other pointing to the sign). The number of ways to choose and arrange these two people is given by:
[tex]\[
\frac{6!}{(6-2)!}
\][/tex]
First, we calculate [tex]\((6-2)!\)[/tex]:
[tex]\[
(6-2)! = 4!
\][/tex]
And,
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
Next, we compute the given expression:
[tex]\[
\frac{6!}{4!} = \frac{720}{24} = 30
\][/tex]
Hence, the number of ways to choose and arrange two people out of six is:
[tex]\[
\boxed{30}
\][/tex]
### Step 3: Choosing 3 people out of 6 to sit in front
Three out of the six people are asked to sit in front of the others. This selection is represented by:
[tex]\[
\frac{6!}{(6-3)! \cdot 3!}
\][/tex]
First, we need to compute both [tex]\((6-3)!\)[/tex] and [tex]\(3!\)[/tex]:
[tex]\((6-3)!\)[/tex] is given by:
[tex]\[
(6-3)! = 3!
\][/tex]
And,
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{6!}{3! \cdot 3!} = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20
\][/tex]
Hence, the number of ways to choose the group of three people to sit in front is:
[tex]\[
\boxed{20}
\][/tex]
