Question

How do you simplify `(7!)/(3!)`?

Answer 1

The result is `840`.

Explanation:

You could use a calculator and actually expand it all out, or you could use the recursive definition of the factorial:

`x! =x*(x-1)!`

For instance, `8! =8*7!`. In our problem, you will see that eventually, things cancel out.

`color(white)=(7!)/(3!)`

`=(7*6!)/(3!)`

`=(7*6*5!)/(3!)`

`qquadqquadqquadvdots`

`=(7*6*5*4*3!)/(3!)`

`=(7*6*5*4*color(red)cancelcolor(black)(3!))/color(red)cancelcolor(black)(3!)`

`=7*6*5*4`

`=42*5*4`

`=210*4`

`=840`

You can use a calculator to check your answer:

![https://www.desmos.com/calculator]()

Hope this helped!

Answer 2

`840`

Explanation:

Recall that `7! =7*6*5*4*3*2*1` and

`3! =3*2*1`

This allows us to rewrite `(7!)/(3!)` as

`(7*6*5*4*3*2*1)/(3*2*1)`

We cancel out common factors on the top and bottom to get

`(7*6*5*4*cancel(3*2*1))/cancel(3*2*1)`

`=>7*6*5*4=840`

Hope this helps!