# Corollaries of Binomial Theorem

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 | |

## Recommended Posts:

- Sum of Binomial coefficients
- Binomial Coefficient | DP-9
- Cauchy's Mean Value Theorem
- Nicomachu's Theorem
- Rosser's Theorem
- Midy's theorem
- Fermat's little theorem
- Fermat's Last Theorem
- Wilson's Theorem
- Sum of product of r and rth Binomial Coefficient (r * nCr)
- Mathematics | PnC and Binomial Coefficients
- Sum of squares of binomial coefficients
- Binomial Random Variables
- Mathematics | Lagrange's Mean Value Theorem
- Extended Midy's theorem

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.