binomial theorem
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:

  1. Each row (or expansion) begins and ends with 1.
  2. Each row is one member longer than the row above it.
  3. Each row has one more term than the power of the binomial.
  4. The sum of the exponents in each term in each row is equal to the power of the binomial.
  5. 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.
  6. Each coefficient is equal to the sum of the two numbers just above it.
  7. 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).

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