 # Binomial Theorem

## A binomial expression has two terms, which are connected by the operators ‘+’ or ‘–’. The total number of terms in the expansion of (a + b)n, for any positive integral n is (n + 1), i.e., one more than the index. In each term of the expansion of (a + b)n, for any positive integral n, the sum of the indices of a and b is same and equal to n. For any positive integer n, the binomial theorem is given by (a + b)n = nc0anb0 + nc1an – 1b1 + nc2an – 2b2 + … + ncna0bn, where nN. In the expansion of (a + b)n , nc0, nc1, nc2, ncn are called the binomial coefficients. The general term in the expansion of (a + b)n is given by (r + 1)th term, i.e., Tr + 1 = ncran – rbr, where 0 ≤ r ≤ n. In case of an even index, the number of the middle terms in the expansion of (a + b)n is one and is given by [(n/2) + 1]th term. In case of an odd index, the number of the middle terms in the expansion of (a + b)n is two which are given by [(n + 1)/2]th term and [(n + 1)/2 + 1]th term.

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• Q1

Find the number of terms in the expansion of [(x + y)3(x - y)3]2.

Marks:1

[(x + y)3(x - y)3]2 = [(x2 - y2)3]2 = (x2 - y2)6 No. of terms in the expansion = n + 1 = 6 + 1 = 7

• Q2

Find the coefficient of x2 in the expansion of (x + 1)6.

Marks:1 • Q3

Find the middle term in the expansion of Marks:1 • Q4

Write the general term for (x - a)n.

Marks:1  