Question

How do you expand `(x + y)^6` using Pascal’s Triangle?

Answer 1

The row of Pascal's triangle starting `1`, `6` gives the sequence of coefficients for the binomial expansion.

Explanation:

Write out Pascal's triangle as far as the row that begins `1`, `6` ...

These are the coefficients you need for the expansion:

`(x+y)^6 = x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6`

Why does this work?

If we write out the value as a product of binomials we have:

`(x+y)^6 = (x+y)(x+y)(x+y)(x+y)(x+y)(x+y)`

If you pick one term from each binomial and multiply them together, then you have made `6` choices of left or right. If you choose all left, then you end up with `x^6` - which you can do just one way. If you choose all right then you end up with `y^6`, which again you can only do one way.

Otherwise, you are making a mixture of left and right choices, analogous to picking your way down from the top of Pascal's triangle to the bottom, via a sequence of left and right branches. The power of `x` resulting is the number of left branches you choose and the power of `y` the number of right branches.

Each number in Pascal's triangle is the sum of the two above, each of which counts the number of ways to reach that point by a sequence of left and right choices. So all of the numbers in Pascal's triangle count the number of ways to reach them by left/right choices starting at the top.