Question

How do you use Pascal's triangle to determine the first four terms of the expanded expression `(x+y)^4`?

Answer 1

By the Binomial Theorem,

`(x+y)^4 = sum_(k=0)^4 ((4),(k)) x^(4-k)y^k`

Pascal's triangle has the property that, if the first row is the `0`-th row and if the first element on a row is also the `0`-th, then the `k`-th on the `n`-th row is

`((n),(k))`, the binomial coefficient.

Pascal's triangle is build by adding the terms from the previous row:

From this diagram, we see that

`(x+y)^4 = color(cyan)1x^4+color(cyan)4x^3y+color(cyan)6x^2y^2+color(cyan)4xy^3+color(cyan)1y^4`