# Corollaries of Binomial Theorem

• Last Updated : 02 Jun, 2018

The expression denotes times.
This can be evaluated as the sum of the terms involving for k = 0 to n, where the first term can be chosen from n places, second term from (n-1) places, term from (n-(k-1)) places and so on. This is expressed as .
The binomial expansion using Combinatorial symbols is • The degree of each term  in the above binomial expansion is of the order n.
• The number of terms in the expansion is n+1.
• Similarly Hence it can be concluded that .

Substituting a = 1 and b = x in the binomial expansion, for any positive integer n we obtain .

Corollary 1: for any non-negative integer n.

Replacing x with 1 in the above binomial expansion, We obtain .

Corollary 2: for any positive integer n.

Replacing x with -1 in the above binomial expansion, We obtain .

Corollary 3:

Replacing x with 2 in the above binomial expansion, we obtain In general, it can be said that Additionally, one can combine corollary 1 and corollary 2 to get another result,  Sum of coefficients of even terms = Sum of coefficients of odd terms.

Since ,

2(   Counting
The coefiecients of the terms in the expansion correspond to the terms of the pascal’s triangle in row n. 1 1         My Personal Notes arrow_drop_up