A rule for writing out the expansion of (a + b)n without performing all the multiplication involved, in which a and b are any real numbers and n is an integer.
When a binomial is raised to whole number powers, the coefficients of the terms form an interesting pattern. Careful examination will reveal that the above expressions exhibit the following features:
- Each row (or expansion) begins and ends with 1.
- Each row is one member longer than the row above it.
- Each row has one more term than the power of the binomial.
- The sum of the exponents in each term in each row is equal to the power of the binomial.
- The coefficients form a symmetrical pattern with respect to its center: every even-numbered row (the very top row is row number 0) has a unique center number while every odd-numbered row has two identical numbers at the center.
- Each coefficient is equal to the sum of the two numbers just above it.
- The sum of the coefficients in each row is equal to 2n, in which n is the row number (set both a and b to 1 and you will see why).