Simplify the factorials first.

\(\displaystyle 3! = 3\cdot2\cdot1 = 6\)

\(\displaystyle 4! = 4\cdot3\cdot2\cdot1 = 24\)

Replace the values back into the problem.

\(\displaystyle \frac{3(2+3!)}{1-4!} = \frac{3(2+6)}{1-24} = \frac{24}{-23}\)

The answer is:  \(\displaystyle -\frac{24}{23}\)

Simplify the denominator.

\(\displaystyle \frac{2(3!+4)}{(2+2)!}=\frac{2(3!+4)}{4!}\)

Rewrite each factorial in the given fraction.  Do not distribute the two through the quantity or this will change the value of the factorial.

\(\displaystyle \frac{2(3!+4)}{4!} =\frac{2([3\times 2\times 1]+4)}{[4\times3\times 2\times 1] } = \frac{2(6+4)}{24}=\frac{2(10)}{24}=\frac{20}{24}\)

Reduce this fraction.

The answer is:  \(\displaystyle \frac{5}{6}\)

To simplify this, we will need to evaluate the parentheses first.  Do not distribute the integer two into the binomial, or the factorial value will change.

\(\displaystyle \frac{2(3+2)!}{3!}=\frac{2(5)!}{3!}\)

Expand the factorials.

\(\displaystyle \frac{2(5)!}{3!} = \frac{2(5\cdot 4\cdot 3\cdot 2\cdot 1)}{3\cdot 2\cdot 1}\)

Simplify the top and bottom terms.

\(\displaystyle 2(5\cdot 4) = 40\)

The answer is:  \(\displaystyle 40\)

Simplify the inner parentheses and expand the factorials.

\(\displaystyle (3!)(3+2)!(\frac{1}{4!}) = (3\times 2\times 1)(5)!(\frac{1}{4 \times 3\times 2\times 1})\)

Cancel out common terms.

\(\displaystyle (5\times 4\times 3\times 2\times 1)(\frac{1}{4}) = 5\times 3\times 2\times 1\)

The answer is:  \(\displaystyle 30\)

In order to solve this, we will need to expand all the factorials.

\(\displaystyle \frac{2!}{3!}\cdot \frac{4!}{6!} = \frac{2\times1}{3\times2\times1}\cdot \frac{4\times3\times2\times1}{6\times5\times4\times3\times2\times1 }\)

Cancel all the common terms and write the remaining numbers.

\(\displaystyle \frac{1}{3 }\cdot \frac{1}{30} = \frac{1}{90}\)

The answer is:  \(\displaystyle \frac{1}{90}\)

In order to simplify the factorials, expand all the terms first.  

\(\displaystyle \frac{3(4!)}{2!+2!+3!} = \frac{3(4\times 3\times2\times 1)}{(2\times 1)+(2\times 1)+(3\times2\times 1)}\)

Simplify the numerator and denominator.

\(\displaystyle \frac{3(24)}{2+2+6} = \frac{72}{10}=\frac{36}{5}\)

The answer is:  \(\displaystyle \frac{36}{5}\)

Evaluate by expanding the factorial in the parentheses.  The zero factorial is a special case which equals to one.

\(\displaystyle 0!\cdot (3+2!)\cdot 0! = 1\cdot (3+(2\times 1))\cdot 1 = 1\cdot (5) \cdot 1\)

The answer is:  \(\displaystyle 5\)

Evaluate by expanding the terms of the factorials.

\(\displaystyle \frac{3!}{5!}\div 3! = \frac{3\times2\times1 }{5\times4\times3\times2\times1 }\div (3\times2\times1 )\)

Cancel out the common terms in the first fraction.

\(\displaystyle \frac{1 }{5\times4 }\div (3\times2\times1 )\)

Change the division sign to a multiplication and take the reciprocal of the second quantity.

\(\displaystyle \frac{1 }{5\times4 }\times \frac{1}{(3\times2\times1 )} =\frac{1}{120}\)

The answer is:  \(\displaystyle \frac{1}{120}\)

Evaluate the terms in the parentheses first.

\(\displaystyle \frac{(5+3)!}{2(2+2)!} =\frac{(8)!}{2(4)!}\)

Expand the factorials.

\(\displaystyle \frac{8\times7\times6\times5\times4\times3\times2\times1 }{2(4\times3\times2\times1 )}\)

The fraction, after simplifying all the terms, becomes:

\(\displaystyle \frac{8\times7\times6\times5}{2} = 4\times7\times6\times5 = 840\)

The answer is:  \(\displaystyle 840\)

Simplify the terms in parentheses first.

\(\displaystyle \frac{3!}{5!} \cdot \frac{(1+5)!}{(3!+1)} = \frac{3!}{5!} \cdot \frac{6!}{(3!+1)}\)

Evaluate each factorial by writing out the terms.

\(\displaystyle \frac{3\times 2\times 1}{5\times 4\times 3\times 2\times 1} \cdot \frac{6\times 5\times 4\times 3\times 2\times 1}{([3\times 2\times 1]+1)}\)

Simplify the parentheses.

\(\displaystyle \frac{3\times 2\times 1}{5\times 4\times 3\times 2\times 1} \cdot \frac{6\times 5\times 4\times 3\times 2\times 1}{7}\)

Simplify all the common terms.

\(\displaystyle \frac{6\times 3\times 2\times 1}{7} = \frac{36}{7}\)

The answer is:  \(\displaystyle \frac{36}{7}\)