16. Binomial Theorem
The Binomial Theorem enables you to expand out expressions like \$(a+b)^n\$ for any positive integer value of n.
There are extensions of this theorem that also allow you to expand out the series for any value of n, but those are outside the scope of the GRE & GMAT.
The Binomial Theorem for Integer Indices states:
\$(x+y)^n=C(n,0)x^n+C(n,1)x^{n-1}y+C(n,2)x^{n-2}y^2+...+C(n,n)y^n\$
Where C(x,y) represents the number of possible combinations of x objects selected y at a time.

A few things worth noting from this theorem:
```1. The terms C(n,0), C(n,1), ..., C(n,n) are called the coefficients.

2. These coefficients for any given n come from the corresponding row of Pascal's Triangle.
Pascal's Triangle is constructed as follows:
1                for \$(x+y)^0=1\$
1 1               for \$(x+y)^1=x+y\$
1 2 1              for \$(x+y)^2=x^2+2xy+y^2\$
1 3 3 1             for \$(x+y)^3=x^3+3x^2y+3xy^2+y^3\$
1 4 6 4 1            for \$(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4\$
... and so on

3. In Pascal's Triangle,
a. each number in every row is equal to the sum of the two numbers immediately above it.
b. the sum of all coefficients in any given row is twice the sum of coefficients in the previous row.
```

E.g. In the expansion of \$(x+y)^10\$, what is the largest coefficient? What is the sum of all coefficients in that expansion?
We see from the above that \$(x+y)^4\$ has coefficients that add up to 16 (=\$2^4\$). So for \$(x+y)^10\$, the sum of coefficients=\$2^10=1024\$.
What is the largest coefficient? From the Pascal's Triangle, we see that center numbers are larger. So for (x+y)^10, this would be C(10,5)=252.