Let's address each part of the problem step by step:
1. Value of [tex]$6!$[/tex]:
The expression [tex]\(6!\)[/tex] (read as "6 factorial") represents the product of all positive integers up to 6. The value of [tex]\(6!\)[/tex] is:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{720}
\][/tex]
2. Number of ways to assign responsibilities to two people:
We need to determine the number of ways to choose 2 people from a group of 6 and assign roles to them. According to the given expression, this can be calculated using the formula:
[tex]\[
\frac{6!}{(6-2)!}
\][/tex]
Here, [tex]\(6!\)[/tex] is the factorial of 6, and [tex]\((6-2)!\)[/tex] is the factorial of 4, because [tex]\(6-2=4\)[/tex].
We already know that [tex]\(6! = 720\)[/tex].
The value of [tex]\(4!\)[/tex] is:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
Therefore, the number of ways:
[tex]\[
\frac{6!}{(6-2)!} = \frac{720}{24} = 30
\][/tex]
So, the number of ways the two people can be chosen and assigned responsibilities is:
[tex]\[
\boxed{30}
\][/tex]
3. Number of ways to choose three people to sit in front:
We need to determine the number of ways to choose 3 people out of 6 to sit in front. According to the given expression, this can be calculated using the combination formula:
[tex]\[
\frac{6!}{(6-3)! \times 3!}
\][/tex]
Here, [tex]\(6!\)[/tex] is the factorial of 6, [tex]\((6-3)!\)[/tex] is the factorial of 3, and [tex]\(3!\)[/tex] is the factorial of 3.
We already know that [tex]\(6! = 720\)[/tex].
The value of [tex]\(3!\)[/tex] is:
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
Therefore, the number of ways:
[tex]\[
\frac{6!}{(6-3)!, 3!} = \frac{720}{6 \times 6} = \frac{720}{36} = 20
\][/tex]
So, the number of ways the group can be chosen is:
[tex]\[
\boxed{20}
\][/tex]
In summary:
- The value of [tex]\(6!\)[/tex] is [tex]\(720\)[/tex].
- The number of ways to assign responsibilities to two people is [tex]\(30\)[/tex].
- The number of ways to choose three people to sit in front is [tex]\(20\)[/tex].
