Binomial coefficient and
NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12
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
T r+1 = n C r a n-r b r
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
n C r = n C n-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 x r in the expansion of (1 + x) n is n C r .
In the binomial expansion (a + b) n , the r th 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 ( n 2 n 2 + 1) th term.
If n is odd, then in the expansion of (a + b) n , the middle terms are ( n + 1 2 n + 1 2 )th term and ( n + 3 2 n + 3 2 )th term.