You already know how to expand a product of brackets by multiplying every term by every
term — see multiplying polynomials.
Expanding (a + b)^2 that way gives:
(a + b)^2 = a^2 + 2ab + b^2
Look only at the numbers in front of each term — the coefficients. They are
1, 2, 1. Expand the next power and the pattern continues:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
The coefficients are now 1, 3, 3, 1. The powers of
a count down while the powers of b
count up (each term's powers always add to n) — that part follows the
. The
interesting question is: where do those coefficients 1, 3, 3, 1 come from?
See it built
Pascal's triangle writes those coefficients out directly. Start with a single
1 at the top. Every row begins and ends with
1, and each number inside is just the sum of the two
numbers above it. Step through it and watch each entry grow from the pair above.
Read off any row to get the coefficients: row 2 is
1, 2, 1 (that's (a+b)^2), and row
3 is 1, 3, 3, 1 (that's
(a+b)^3). The triangle's diagonals are themselves famous
.