# What is (5!3!)/(6!)?

Apr 30, 2018

$1$

#### Explanation:

This problem can be made easier by rewriting the equation:

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

We can cancel quite a few numbers:

(cancel(5 * 4 * 3 * 2 * 1) * 3 * 2 * 1)/( 6 * cancel(5 * 4 * 3 * 2 * 1)

$\frac{3 \cdot 2 \cdot 1}{6}$

$\frac{6}{6} = 1$

Apr 30, 2018

The answer is $1$.

#### Explanation:

The ! is a factorial, which means if you have, for example, 4!, you just do $4 \cdot 3 \cdot 2 \cdot 1 = 24$.

Method 1:

Multiply the 6! out to be 6*5! and get (5!3!)/(6*5!).
(We do this so we can cancel out the 5!s in the next step.)
Cancel out the 5!s and get: (3!)/6
Now just multiply out the 3! to be $3 \cdot 2 \cdot 1 = 6$.
You end up with $\frac{6}{6}$, which equals $1$.

This looks like a lot, but it's actually quite nice because you don't have to multiply out the 5! or 6! completely.

Method 2:

Another way to do this is just completely multiply everything out like this:
$\frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 3 \cdot 2 \cdot 1}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$
Cancel everything out that you can, and you should end up with the same answer, $1$.