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Binomial coefficient and

NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Binomial Expression

An expression consisting of two terms, connected by + or – sign is called binomial expression.

Binomial Theorem

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.