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.
Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.