Example: 1  
Find the middle term(s) in the expansion of

Solution: 1  
Since , there will be terms in the expansion, which means that there will be middle terms in the expansion, the and the :

Example: 2  
Is there any term in the expansion of that will be independent of ?

Solution: 2  
The general term in the expansion is
Thus, for the term that is independent of , we have
Thus, the term free of is the term given by

Example: 3  
Evaluate the sum .

Solution: 3  
We have already evaluated this sum in the chapter on & . That approach was as follows: this sum basically counts the number of all subgroups of a set of size ; this can also be counted by focusing on each element of the set, which has two corresponding choices – you either include it into your subgroup or you don’t, which means that the total number of ways to form subgroups is . The sum of the binomial coefficients therefore equals .
Here, we evaluate the same sum using a binomial approach. Consider the following expansion:
If we put , we simply obtain
Thus, the same result is obtainable from both a combinatorial and a binomial approach.
We can also derive another useful result by putting in the above relation, so that we obtain
This states the sum of the evennumbered coefficients is equal to the sum of the oddnumbered coefficients. Can you prove this using a combinatorial approach?
As an exercise, prove the following relations:
