How do I find the constant term of a binomial expansion?

The expansion of a binomial is given by :

##(x+y)^n=( (n), (0) )*x^n+( (n), (1) )*x^(n-1)*y^1+...+( (n), (k) )*x^(n-k)*y^k+...+( (n), (n) )*y^n = sum_(k=0)^n*( (n), (k) )*x^(n-k)*y^k ## where ##x, y in RR##, ##k, n in NN##, and ##( (n), (k) )## denotes combinations of ##n## things taken ##k## at a time.

## ( (n), (k) )*x^(n-k)*y^k ## is the general term of the binomial expansion.

We also have the formula: ##( (n), (k) )=(n!)/(k!*(n-k)!)##, where ##k! = 1*2*...*k##

We have three cases:

Case 1: If the terms of the binomial are a variable and a constant ##(y=c##, where ##c## is a constant), we have ##(x+c)^n=( (n), (0) )*x^n+( (n), (1) )*x^(n-1)*c^1+...+( (n), (k) )*x^(n-k)*c^k+...+( (n), (n) )*c^n ##

We can see that the constant term is the last one: ##( (n), (n) )*c^n## (as ##( (n), (n) )## and ##c^n## are constant, their product is also a constant).

Case 2: If the terms of the binomial are a variable and a ratio of that variable (##y=c/x##, where ##c## is a constant), we have: ## (x+c/x)^n=( (n), (0) )*x^n + ( (n), (1) )*x^(n-1)*(c/x)^1+...+( (n), (k) )*x^(n-k)*(c/x)^k+...+( (n), (n) )*(c/x)^n ##

This time, we see that the constant term is not to be found at the extremities of the binomial expansion. So, we should have a look at the general term and try to find out when it becomes a constant: ## ( (n), (k) )*x^(n-k)*(c/x)^k=( (n), (k) )*x^(n-k)*c^k*1/x^k = (( (n), (n) )*c^k)*(x^(n-k))/x^k = (( (n), (k) )*c^k)*x^(n-2k) ##.

We can see that the general term becomes constant when the exponent of variable ##x## is ##0##. Therefore, the condition for the constant term is: ##n-2k=0 rArr## ##k=n/2## . In other words, in this case, the constant term is the middle one (##k=n/2##).

Case 3: If the terms of the binomial are two distinct variables ##x## and ##y##, such that ##y## cannot be expressed as a ratio of ##x##, then there is no constant term . This is the general case ##(x+y)^n##