Sure, let's evaluate these expressions step-by-step.
1. Calculating [tex]\( 6! \)[/tex]:
The factorial of a number [tex]\( n \)[/tex], denoted as [tex]\( n! \)[/tex], is the product of all positive integers less than or equal to [tex]\( n \)[/tex].
So, for [tex]\( 6! \)[/tex]:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Thus, [tex]\( 6! = 720 \)[/tex].
2. Calculating [tex]\( 3! \cdot 2! \)[/tex]:
First, we need to find the factorials of 3 and 2.
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
[tex]\[
2! = 2 \times 1 = 2
\][/tex]
Now, multiply these two results:
[tex]\[
3! \cdot 2! = 6 \times 2 = 12
\][/tex]
Thus, [tex]\( 3! \cdot 2! = 12 \)[/tex].
3. Calculating [tex]\( \frac{61}{31} \)[/tex]:
Here, we simply divide 61 by 31. This gives:
[tex]\[
\frac{61}{31} \approx 1.967741935483871
\][/tex]
Thus, [tex]\( \frac{61}{31} \approx 1.967741935483871 \)[/tex].
In summary, the evaluated expressions are:
[tex]\[
6! = 720
\][/tex]
[tex]\[
3! \cdot 2! = 12
\][/tex]
[tex]\[
\frac{61}{31} \approx 1.967741935483871
\][/tex]
