The binomial theorem expands an integer power of a two-term sum without repeatedly multiplying every factor. Each term combines a binomial coefficient with powers of the two summands. Knowing that pattern also lets you find one term without writing the entire expansion.
What does a binomial coefficient count?
The coefficient with upper entry n and lower entry k counts selections of k positions from n. It is not an ordinary fraction: the stacked entries identify a combination. Here n is a nonnegative integer and k ranges from zero to n; zero factorial equals one.
Selecting two positions from eight or leaving two out of eight gives the same count as selecting six. This explains the symmetry between lower entries k and n − k in a fixed row.
Expanding a sum with the binomial theorem
The finite formula below assumes a nonnegative integer exponent n. In each successive term, the exponent of a decreases by one while the exponent of b increases by one. Their sum remains n. There are n + 1 positions in the expansion before like terms are combined.
For a difference, include the negative sign in the second summand. Odd powers of that negative summand produce negative terms, while even powers produce positive ones. The finite formula should not be applied unchanged to arbitrary real exponents.
Reading Pascal’s triangle from row zero
The top 1 is row zero. Each following row begins and ends with 1, and an interior entry is the sum of the two entries immediately above it. Row three is therefore 1, 3, 3, 1, the coefficients for a cube.
Worked expansion: the cube of x + 3
Use row three, lower the exponent of x from three to zero, and increase the exponent of 3 from zero to three. Multiply each coefficient by its numerical power before collecting the result. The middle terms do not disappear.
Finding the fifth term without expanding everything
The general term is numbered k + 1 because the first term uses k = 0. Thus the fifth term uses k = 4, not k = 5.
For the fifth term of the fifth power of the sum of the cube of x and 2, take a as the cube of x, b as 2, and n as 5. The formula gives the following result when terms are arranged in descending powers of x.
For the counting interpretation behind the coefficients, see permutations and combinations.
