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 [Tex]b^{n-k} [/Tex]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, [Tex]^nC_0 + ^nC_2 + .. = ^nC_1 + ^nC_3 + … [/Tex]Sum of coefficients of even terms = Sum of coefficients of odd terms. Since , 2([Tex]^nC_0 + ^nC_2 + .. = 2^{n-1} [/Tex]

Counting The coefficients of the terms in the expansion correspond to the terms of the pascal’s triangle in row n.