Table of binomial coefficients

The following is the expansion of the expression (x + y)n , where n is a non-negative integer. It can be shown by the ordinary multiplication that: Notice that the coefficients of terms in the above expansions form what is known as the Pascal's triangle.

Each row of Pascal's triangle starts and ends with a 1; others can be obtained by adding the two terms on either side of it in the preceding row.

Note also that the above Pascal's triangle can be written as: In general, if n is a positive integer, the expansion of (x + y)n is given by the following: This is called the Binomial Theorem for any positive integral value of n.

Note The expansion of (x + y)n consists of (n + 1) terms of which is the (r + 1)th term.

If we write -y in place of y, we get the following: The terms are alternately positive and negative.

Example Expand ( 2 x + 3 y )5 and simplify each term.