The binomial expansion is easier if you don't call the first
term "the first term" but instead call it "the zero-th term",
and call the second term "the first term" and then call the last
term the nth term, (which is really the (n+1)st term).

If you use that "start counting from 0 instead of 1" convention 
then the Rth term of 

is  which is sometimes written  and sometimes
written  but it always means , 

the 0th term's coefficient is  or  or 

which means 

The 1st term's coefficient is  or  or 

which means  

=  =

=  =



The 2nd term's coefficient is  or  or 

which means  

=  =

=  =



You can also just skip all that cancelling and observe
the pattern of what's always left after cancelling:

If you needed the 3rd term's coefficient (really the 4th term's), 
it would be

 

If you needed the 4th term's coefficient (really the 5th term), it would be

 

If you needed the 5th term's coefficient (really the 6th term), it would be



etc., etc.

So to get the numerator of any term's coefficient, you start with
the outer exponent, and multiply it by 1 less, then 1 less, etc.
until you have as many factors as the number of term (starting
counting from zero, not 1). Then the denominator is just that
number factorial.  There are always as many factors on top
as the number of term (starting counting form 0, not 1). And
of course the denominator has that same number of factors, too.

Edwin