An expression consisting of two terms, connected by + or – sign is called binomial expression.
If a and b are real numbers and n is a positive integer, then
The general term of (r + 1)th term in the expression is given by
Tr+1 = nCr an-r br
Some Important Observations from the Binomial Theorem
The total number of terms in the binomial expansion of (a + b)n is n + 1.
The sum of the indices of a and b in each term is n.
The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficient and
nCr = nCn-r, r = 0, 1, 2, 3,…, n
The values of the binomial coefficient steadily increase to a maximum and then steadily decrease.
The coefficient of xr in the expansion of (1 + x)n is nCr.
In the binomial expansion (a + b)n, the rth term from the end is (n – r + 2)th term from the beginning.
Middle Term in the Expansion of (a + b)n
If n is even, then in the expansion of (a + b)n, the middle term is ( + 1) th term.
If n is odd, then in the expansion of (a + b)n, the middle terms are ()th term and ()th term.