1.1 Basics of the Binomial Theorem
\((x+y)^n=\sum\limits_{r=0}^n\ C(n,r)x^ry^{n-r}\), where \(n\in \mathbb{N}\). This is what we call the Binomial Theorem, and holds for all reals \(x,y\).
In the above theorem, if we put \(x=y=1\), we arrive at the following identity:
\(2^n=\sum\limits_{r=0}^nC(n,r)\). Combinatorially speaking, out of a set of \(n\) elements, the total number of ways that a selection can be formed containing any number of elements is \(2^n\).
Also, we know exactly the coefficient of any power of the variables from this formula. For example, the coefficient of \(x^r\) in \((x+a)^n\) would be \(C(n,r)a^{n-r}\). Usually, we will denote the rth term of a binomial expansion\((x+y)^n\) as \(T_r=C(n,r-1)x^{r-1}y^{n-r+1}\).
Test your concepts:
1. Compute \((x+y)^n+(x-y)^n\).
2. Compute \((x+y)^n-(x-y)^n\).
Note that in the expansion of \((x+y)^n\), there are \(n+1\) terms altogether.
1.2 Middle and Greatest Terms
The middle term depends on the parity of \(n\). Let’s consider the expansion \((x+y)^n\)
a) When \(n\) is even, the middle term is simply the \(\left( \frac{n}{2}+1\right)\)th term, i.e \(C(n,\frac{n}{2})x^{\frac{n}{2}}y^{\frac{n}{2}}\).
b) When \(n\) is odd, we have 2 middle terms, i.e \(\frac{n+1}{2}\)th and \(\frac{n+3}{2}\)th terms form the middle terms. Thus if \(n=5\), then the 3rd and 4th terms form the middle terms.
Now let’s calculate the greatest term of a binomial expansion\((1+x)^n\).
\(\frac{T_{r+1}}{T_r}=\frac{C(n,r)x^r}{C(n,r-1)x^{r-1}}=\frac{n-r+1}{r}x\).
If \(T_{r+1}\ge T_r\Rightarrow \frac{n-r+1}{r}|x|\ge 1\Rightarrow r\le \frac{|x|(n+1)}{|x|+1}\).
Thus, if \(m=\frac{|x|(n+1)}{|x|+1}\in \mathbb{Z}\), then \(T_m=T_{m+1}\) are the greatest terms.
Else \(T_{[m]+1}\) is the greatest term, where \([]\) is the g.i.f.
In similar ways, we have the greatest coefficient of a binomial expansion as
a) if \(n=2k\) is even, then greatest coefficient is \(C(n,k)\)
b) if \(n=2k+1\) is odd, then greatest coefficient is \(C(n,k),C(n,k+1)\).