Evaluate (8!)/(2!2!2!)

$\frac{8!}{2! 2! 2!}$

Step 1

Multiply $2!$ by $2!$ .

$\frac{8!}{{2!}^{2} 2!}$

Step 2

Multiply ${2!}^{2}$ by $2!$ by adding the exponents.

Multiply ${2!}^{2}$ by $2!$ .

Raise $2!$ to the power of $1$ .

$\frac{8!}{{2!}^{2} {2!}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8!}{{2!}^{2 + 1}}$

Add $2$ and $1$ .

$\frac{8!}{{2!}^{3}}$

Step 3

Simplify the numerator.

Expand $8!$ to $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

Multiply $8$ by $7$ .

$\frac{56 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $56$ by $6$ .

$\frac{336 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $336$ by $5$ .

$\frac{1680 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $1680$ by $4$ .

$\frac{6720 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $6720$ by $3$ .

$\frac{20160 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $20160$ by $2$ .

$\frac{40320 \cdot 1}{{2!}^{3}}$

Multiply $40320$ by $1$ .

$\frac{40320}{{2!}^{3}}$

Step 4

Simplify the denominator.

Expand $2!$ to $2 \cdot 1$ .

$\frac{40320}{{(2 \cdot 1)}^{3}}$

Multiply $2$ by $1$ .

$\frac{40320}{2^{3}}$

Raise $2$ to the power of $3$ .

$\frac{40320}{8}$

Step 5

Divide $40320$ by $8$ .

$5040$