Binomial Theorem


The binomial theorem formula is given in order to make it easy to expand the complex values that have powers. It is based on the arrangement of Pascal’s triangle that means all the values of the expressed theorem can be arranged to form the Pascal’s triangle. The Binomial Theorem can be described as the algebraic expansion of powers. In elementary algebra according to the theorem it is possible to expand the power (p + q)n into a sum of the form

(p +q)n = (n0) pnq0 + (n1) pn-1q1 + (n2) pn-2q2 +…….+(nn-1) p1qn-1 + (nn) p0qn,

Here each (nk) is the positive integer and called as binomial coefficient. The formula is also called as the binomial formula or the binomial identity.

And using summation notation it can be written as:

(p + q)n = ånk=0 (n) pn-k qk = ånk=0 (nk) pkqn-k.

And in order to the variant of the binomial formula we have to put all the (y) = 1 that means we have to substitute one in place of ‘y’, so there will be only one variable in the expression that gives.

(1 + p)n = (n0)p0 + (n1)p1 + (n2)p2 +………..+(nn-1)pn-1 + (nn) pn,

Or   (1 + p)n =  = ånk=0 (nk)pk.

Now if we talk about the binomial coefficient then it can be explained as the coefficients that appear in the binomial expansion are called binomial coefficients and these are written as (nk) and normally pronounced as “n choose k”.

The coefficient of pn-k qk is given by the formula

(nk) = n! /  [k! (n - k)!], that can be defined in terms of factorial functions n!,

(nk) = [n(n-1)……(n – k +1)] / [k(k-1)…..1] = Pkl=1 [(n – l + 1) / l],

In the general sense if we try to expand the term (p + q)n then it will produce the sum of the 2n products of the form z1z2……..zn, where each  zi is ‘p’ or ‘q’ and if we try to rearrange these factors we gets that each products equal to pn-kqk, for some ‘k’ between 0 and n.

For a given ‘k’, we can conclude the following points.

1.       The total number of copies of pn-k yk in the expansion.

2.       The total number of n- characters having x, y strings in the exact ‘k’ positions.

3.       The number of ‘k’ elements are the subset of 1,2,3…….,n.

Math Topics