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.
Hence it can be concluded that .
Substituting a = 1 and b = x in the binomial expansion, for any positive integer n we obtain
for any non-negative integer n.
Replacing x with 1 in the above binomial expansion, We obtain
for any positive integer n.
Replacing x with -1 in the above binomial expansion, We obtain
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.
The coefiecients of the terms in the expansion correspond to the terms of the pascal’s triangle in row n.