Pascal's triangle

There’s another way of finding the coefficients in a binomial expansion: it works well for small values of but takes a long time for larger values of .

The coefficients in the expansion of correspond to the rows of Pascal’s triangle:

row |
  0 |              1
  1 |            1   1
  2 |          1   2   1
  3 |        1   3   3   1
  4 |      1   4   6   4   1
  5 |    1   5  10  10   5   1
  6 |  1   6  15  20  15   6   1
       ^                   ^
       |_ column 0         |_ column 5

For example then, the coefficients of the expansion of are , , , , and because they correspond to row of Pascal’s triangle.

Constructing Pascal’s triangle

We just put a at the top, then we construct the next row by adding the two numbers above it.

For example, to get the in the 4th row, we add the and the above it. To get the in the 5th row, we add the and the above it.

Common binomial values

If you look at the common binomial expansions, you’ll see this pattern more clearly!