Not all questions can be subjected to the method(s) described earlier. For example, consider the sum given by
Let us first go through a combinatorial approach, using the observation that and so on, so that can be rewritten as
Consider a general term of this sum, which is of the form . We can think of this as the number of ways of selecting boys from a group of boys and girls from a group of girls. The total number of people we are thus selecting is . Therefore, represents the total number of ways of selecting people out of a group of , so that is simply .
Now to a binomial approach. This will involve generating the general term somehow, which is the same as . Consider the general expansion of .
We have to have the terms and so on, which suggests that we write twice, but in the second expansion we reverse the terms, multiply, and see what terms contain the (combinations of) coefficients we require.
Multiplying, we find on the left hand side we have , while on the right hand side, the terms containing the (combinations of) coefficients we want will always be of the form , that is, the power of will be . No other terms will contain , verify this for yourself. Thus, the sum is actually the total coefficient of on the right hand side, and from the left hand side we know that the coefficient of would be simply . Thus,
A very similar approach could have been
Coefficient of in  
Coefficient of in  
Coefficient of in 
Example: 9  
Find the sum given by 
Solution: 9  
Note that can be rewritten as
Using a combinatorial approach, the sum should be immediately obvious to the alert reader as . In brief, this is because the right hand side represents, as an example, the total number of ways of selecting people from a group of boys and girls, etc.
Now, we discuss the binomial expansion approach:
Thus, we observe that the required sum is the coefficient of in , which is simply .

Example: 10  
Find the sum given by 
Solution: 10  
We have already evaluated this sum in ; here we’ll use a binomial approach. Note that
Thus,

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