Evaluate (9!)/(2!2!2!)

$\frac{9!}{2! 2! 2!}$

Step 1

Multiply $2!$ by $2!$ .

$\frac{9!}{{2!}^{2} 2!}$

Step 2

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

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

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

$\frac{9!}{{2!}^{2} {2!}^{1}}$

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

$\frac{9!}{{2!}^{2 + 1}}$

Add $2$ and $1$ .

$\frac{9!}{{2!}^{3}}$

Step 3

Simplify the numerator.

Expand $9!$ to $9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

Multiply $9$ by $8$ .

$\frac{72 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $72$ by $7$ .

$\frac{504 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $504$ by $6$ .

$\frac{3024 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $3024$ by $5$ .

$\frac{15120 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $15120$ by $4$ .

$\frac{60480 \cdot 3 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $60480$ by $3$ .

$\frac{181440 \cdot 2 \cdot 1}{{2!}^{3}}$

Multiply $181440$ by $2$ .

$\frac{362880 \cdot 1}{{2!}^{3}}$

Multiply $362880$ by $1$ .

$\frac{362880}{{2!}^{3}}$

Step 4

Simplify the denominator.

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

$\frac{362880}{{(2 \cdot 1)}^{3}}$

Multiply $2$ by $1$ .

$\frac{362880}{2^{3}}$

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

$\frac{362880}{8}$

Step 5

Divide $362880$ by $8$ .

$45360$