Corollaries of Binomial Theorem

Last Updated : 02 Jun, 2018

The expression $(a+b)^n$ denotes $(a+b)(a+b)(a+b) ... n$ times.
This can be evaluated as the sum of the terms involving $a^k b^{n-k}$ for k = 0 to n, where the first term can be chosen from n places, second term from (n-1) places, $k^{th}$ term from (n-(k-1)) places and so on. This is expressed as $(a+b)^n = \sum\limits_{k=0}^n ^nC_k a^{n-k} b^k$ .
The binomial expansion using Combinatorial symbols is

$(a+b)^n = ^nC_0 a^n b^0 + ^nC_1 a^{n-1} b^1 + ^nC_2 a^{n-2} b^2 .. + ^nC_{n-k} a^k b^{n-k} .. +^nC_n a^0 b^n$

The degree of each term $a^k$ $b^{n-k}$ in the above binomial expansion is of the order n.
The number of terms in the expansion is n+1.
$^nC_k = n!/k!(n-k)!$
Similarly $^nC_{n-k} = n!/(n-k)!(n-(n-k))! = n!/(n-k)!k!$
Hence it can be concluded that $^nC_k = ^nC_{n-k}$ .

Substituting a = 1 and b = x in the binomial expansion, for any positive integer n we obtain
$(1+x)^n = ^nC_0 + ^nC_1 x^1 + ^nC_2 x^2 ..+ ^nC_n x^n$ .

Corollary 1:

$\sum\limits_{k=0}^n ^nC_k = 2^n$

for any non-negative integer n.

Replacing x with 1 in the above binomial expansion, We obtain
$^nC_0 + ^nC_1 + ^nC_2 .. + ^nC_n = (1+1)^n = 2^n$ .

Corollary 2:

$\sum\limits_{k=0}^n ^nC_k = 0$

for any positive integer n.

Replacing x with -1 in the above binomial expansion, We obtain
$^nC_0 + ^nC_1 (-1) + ^nC_2 (-1)^2 .. + ^nC_n (-1)^n = (1+(-1))^n = 0$ .

Corollary 3:

Replacing x with 2 in the above binomial expansion, we obtain
$^nC_0 + ^nC_1 2 + ^nC_2 2^2 .. + ^nC_n 2^n = (1+2)^n = 3^n$

In general, it can be said that

$\sum\limits_{k=0}^n (2^k) ^nC_k = 3^n$

Additionally, one can combine corollary 1 and corollary 2 to get another result,

$^nC_0 + ^nC_1 (-1) + ^nC_2 (-1)^2 .. + ^nC_n (-1)^n = (1+(-1))^n = 0$

$^nC_0 + ^nC_2 + .. = ^nC_1 + ^nC_3 + ...$
Sum of coefficients of even terms = Sum of coefficients of odd terms.

Since $\sum\limits_{k=0}^n ^nC_k = 2^n$ ,

2( $^nC_0 + ^nC_2 + ..) = 2^n$

$^nC_0 + ^nC_2 + .. = 2^{n-1}$

$^nC_0 + ^nC_2 + .. = ^nC_1 + ^nC_3 + .. = 2^{n-1}$

Counting
The coefiecients of the terms in the expansion $(a+b)^n$ correspond to the terms of the pascal’s triangle in row n.

$(a+b)^0$	1	1
$(a+b)^1$	$a+b$	$1 \ 1$
$(a+b)^2$	$a^2+2ab+b^2$	$1 \ 2 \ 1$
$(a+b)^3$	$a^3+3a^2b+3ab^2+b^3$	$1 \ 3 \ 3 \ 1$

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