Let's evaluate the given expressions step-by-step.
1. Evaluating [tex]\(6!\)[/tex]:
The exclamation mark "!" denotes a factorial. For any positive integer [tex]\( n \)[/tex], [tex]\( n! \)[/tex] is calculated as the product of all positive integers from 1 to [tex]\( n \)[/tex]. So:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Therefore:
[tex]\[
6! = 720
\][/tex]
[tex]\[
6! = \boxed{720}
\][/tex]
2. Evaluating [tex]\(3! \cdot 2!\)[/tex]:
Firstly, calculate the factorial of 3 and 2 separately:
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
[tex]\[
2! = 2 \times 1 = 2
\][/tex]
Now, multiply these two results together:
[tex]\[
3! \cdot 2! = 6 \times 2 = 12
\][/tex]
Therefore:
[tex]\[
3! \cdot 2! = \boxed{12}
\][/tex]
3. Evaluating [tex]\(\frac{6!}{3!}\)[/tex]:
To evaluate this, we use the numbers we already calculated:
[tex]\[
6! = 720
\][/tex]
[tex]\[
3! = 6
\][/tex]
Now, compute the division:
[tex]\[
\frac{6!}{3!} = \frac{720}{6} = 120
\][/tex]
Therefore:
[tex]\[
\frac{6!}{3!} = \boxed{120}
\][/tex]
In summary, the evaluated expressions are:
[tex]\[
6! = \boxed{720}
\][/tex]
[tex]\[
3! \cdot 2! = \boxed{12}
\][/tex]
[tex]\[
\frac{6!}{3!} = \boxed{120}
\][/tex]
