We can pick the 1st person as any of the 5 persons.
That's 5 ways to pick the first person.

For each of those 5 ways to pick the 1st person,
we can pick the 2nd person as any of the 4 remaining
unpicked persons.
That's 5*4 or 20 ways to pick the first two people.

For each of those 5*4 or 20 ways to pick the first two
persons,
we can pick the 3rd person as any of the 3 remaining
unpicked persons.
That's 5*4*3 or 60 ways to pick the first three people. 

For each of those 5*4*3 or 60 ways to pick the first three
persons,
we can pick the 4th person as any of the 2 remaining
unpicked persons.
That's 5*4*3*2 or 120 ways to pick the first four people.

For each of those 5*4*3*2 or 120 ways to pick the first four
persons,
we can only pick the 5th person as the 1 remaining
unpicked person.
That's 5*4*3*2*1 or 120 ways to pick the five people.

Notice that 5*4*3*2*1 = 120 = 5!, read "five factorial".

Edwin