Question

a coin is tossed five times in how many ways can one get exactly no heads? five heads?

Answer 1

One can get no heads in exactly one way. We can see this as (5C0)=1

One can get 5 heads in exactly one way. We can see this as (5C5)=1

Answer 2

in how many ways can one get exactly one head? four heads?

Answer 3

Well, lets enumerate them:
For one head:
HTTTT, THTTT, TTHTT, TTTHT, TTTTH

i.e. 5 ways

This is equal to (5C1), that is the binomial coefficient or "choose" function. Now you try with 4 heads.

Answer 4

5c2