Permutation

₆P₄ is,
start from 6 and multiply 4 numbers,
6⋅5⋅4⋅3.

6⋅5 = 30
4⋅3 = 12

30⋅12 = 360

So 360 is the answer.

_nP_r is used
when choosing r things from n things
and arranging them in a row.

It says
choose 3 students from 9 students
and arrange them in a row.

So the number of the ways is
₉P₃.

₉P₃ is,
start from 9 and multiply 3 numbers,
9⋅8⋅7.

9⋅8 = 72

72⋅7 = 504

So 504 is the answer.

There are 8 numbers.

To make a 4-digit number,
choose 4 numbers
and arrange them in a row.

So the number of the ways is
₈P₄.

₈P₄ is,
start from 8 and multiply 4 numbers,
8⋅7⋅6⋅5.

8⋅7 = 56
6⋅5 = 30

56⋅30 = 1680

So 1680 is the answer.

There are 6 numbers.

To make a 3-digit number,
choose 3 numbers
and arrange them in a row.

The number of the ways is
₆P₃.

But there's a case to subtract.

If 0 is in the hundreds,
then it's not a 3-digit number:
[012] is not a 3-digit number.

So you should subtract this case.

If 0 is in the hundreads,
there are 5 numbers to choose,
choose 2 numbers,
and arrange them in tens and ones.

The number of the ways to subtract is
₅P₂.

So the number of ways
to make a 3-digit number is
₆P₃ - ₅P₂.

₆P₃ is,
start from 6 and multiply 3 numbers,
6⋅5⋅4.

Minus,
₅P₂ is,
start from 5 and multiply 2 numbers,
5⋅4.

So ₆P₃ - ₅P₂
= 6⋅5⋅4 - 5⋅4.

6⋅5⋅4 - 5⋅4
= 5⋅4(6 - 1)

Common Monomial Factor

5⋅4 = 20
6 - 1 = 5

20⋅5 = 100

So 100 is the answer.