Pascal’s triangle

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

The coefficients in the expansion of (a+b)^n 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 (a+b)^5 are 1, 5, 10, 10, 5 and 1 because they correspond to row 5 of Pascal’s triangle.

Constructing Pascal’s triangle

We just put a 1 at the top, then we construct the next row by adding the two numbers above it.

For example, to get the 6 in the 4th row, we add the 3 and the 3 above it. To get the 10 in the 5th row, we add the 4 and the 6 above it.

Common binomial values

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

flashcards

QuestionAnswer
row 0 of Pascal’s triangle1
row 1 of Pascal’s triangle1 1
row 2 of Pascal’s triangle1 2 1
row 3 of Pascal’s triangle1 3 3 1
row 4 of Pascal’s triangle1 4 6 4 1
row 5 of Pascal’s triangle1 5 10 10 5 1
row 6 of Pascal’s triangle1 6 15 20 15 6 1
How are coefficients in (a+b)^n related to Pascal’s triangle?They correspond to the values in row n of Pascal’s triangle.
What are the coefficients of (a+b)^5 according to Pascal’s triangle?1, 5, 10, 10, 5, 1 (row 5).
How do you construct the next row in Pascal’s triangle?Add the two numbers directly above it.
How is the 6 in row 4 of Pascal’s triangle obtained?By adding the two 3s above it.
How is the 10 in row 5 of Pascal’s triangle obtained?By adding the 4 and the 6 above it.
What is the column 0 number in every row of Pascal’s triangle?1.
What is column n number in row n of Pascal’s triangle?1.
What is a disadvantage of using Pascal’s triangle for binomial expansion?It works well for small n but takes a long time for larger n.