Factorial

n! means
multiply from n to 1.

n! = n⋅(n - 1)⋅(n - 2) ... 3⋅2⋅1

[ ! ] is read as [factorial].

Special values:
1! = 1
0! = 1

5! is,
multiply from 5 to 1,
5⋅4⋅3⋅2⋅1.

5⋅4 = 20
3⋅2⋅1 = 6

20⋅6 = 120

So 120 is the answer.

The denominator is 4!.

So change 7! to 7⋅6⋅5⋅4!.

(7! = 7⋅6⋅5⋅4⋅3⋅2⋅1
= 7⋅6⋅5⋅4!)

Cancel both 4!.

6⋅5 = 30

7⋅30 = 210

So 210 is the answer.

There are 3 letters:
a, b, c.

And there are 4 numbers:
1, 2, 3, 4.

It says
all letters are adjacent to each other.
So group the letters.

So there are,
[group], 1, 2, 3, 4,
5 things to arrange.

n! is used
when finding the number of ways
to arrange n things.

So the number of ways
to arrange these 5 things is
5!.

For each case,
the letters [a, b, c] are also arranged.

So multiply the number of ways
to arrange these 3 letters:
3!.

Rule of Product

So the number of the ways is
5!⋅3!.

5! is,
multiply from 5 to 1,
5⋅4⋅3⋅2⋅1.

3! is,
multiply from 3 to 1,
3⋅2⋅1.

5⋅4 = 20
3⋅2⋅1 = 6

3⋅2⋅1 = 6

6⋅6 = 36

20⋅36 = 720

So 720 is the answer.