Explanation:
Binomial
theorem
The binomial coefficients appear as the entries of Pascal's triangle
The binomial theorem describes the algebraic
expansion of powers of a binomial. According to the theorem, it is possible to expand the
power (x + y)^{n} into a sum involving terms of the form ax^{b}y^{c}, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the
corresponding power is usually omitted from the term. For example,
The
coefficient a in the term of x^{b}y^{c} is known as the binomial coefficient or (the two have the same value).
These coefficients for varying n and b can be arranged to form Pascal's triangle. These
numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set.
Statement of the theorem
According to
the theorem, it is possible to expand any power of x + y into a sum of the form
where each is a specific positive
integer known as binomial coefficient. This formula is also referred to as the Binomial Formula or the Binomial Identity. Using summation notation, it can be
written as
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Binomial_theorem
The
above explanation is copied from Wikipedia, the free encyclopedia and is
remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported
License.
(Our solved example in mathguru.com also uses the below concept. This
is our own explanation, it is not taken from Wikipedia.)
MIDDLE TERM OF THE BINOMIAL
EXPANSION
If we have
to find the middle term of an expansion it depends upon the number of terms in
it. The number of terms in the expansion of (a + b)^{ n} depends on the
index "n". The index "n" is either even or odd.
If n is odd
Let n = 2k+1
The number of terms is n+1 i.e., (2k + 1) + 1 = 2k + 2.
In this case, there are two middle terms and are after k terms.