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Evaluate each expression.

[tex] 6! = \square [/tex]

[tex] 3! \cdot 2! = \square [/tex]

[tex] \frac{6!}{3!} = \square [/tex]


Sagot :

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]