Let's evaluate each of the given expressions step-by-step.
### 1. Evaluating [tex]\( 6! \)[/tex]
The factorial of a number [tex]\( n \)[/tex] is the product of all positive integers from 1 to [tex]\( n \)[/tex]. So, for [tex]\( 6! \)[/tex], we have:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Therefore,
[tex]\[
6! = 720
\][/tex]
### 2. Evaluating [tex]\( 3! \cdot 2! \)[/tex]
First, we find [tex]\( 3! \)[/tex] and [tex]\( 2! \)[/tex]:
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
[tex]\[
2! = 2 \times 1 = 2
\][/tex]
Now, multiply these two results together:
[tex]\[
3! \cdot 2! = 6 \cdot 2 = 12
\][/tex]
Therefore,
[tex]\[
3! \cdot 2! = 12
\][/tex]
### 3. Evaluating [tex]\( \frac{6!}{3!} \)[/tex]
We already know [tex]\( 6! \)[/tex] and [tex]\( 3! \)[/tex]:
[tex]\[
6! = 720
\][/tex]
[tex]\[
3! = 6
\][/tex]
Now, divide [tex]\( 6! \)[/tex] by [tex]\( 3! \)[/tex]:
[tex]\[
\frac{6!}{3!} = \frac{720}{6} = 120
\][/tex]
Therefore,
[tex]\[
\frac{6!}{3!} = 120
\][/tex]
### Final Results
So, the evaluated expressions are as follows:
[tex]\[
6! = 720
\][/tex]
[tex]\[
3! \cdot 2! = 12
\][/tex]
[tex]\[
\frac{6!}{3!} = 120
\][/tex]
