Question

How do you use pascals triangle to expand `(2x-y)^3`?

Answer 1

`8x^3 - 12x^2y + 6xy^2 - y^3`

Explanation:

In general, for `(a+b)^k`, the expansion is `C(k,0)a^kb^0+C(k,1)a^(k−1)b^1+C(k,2)a^(k−2)b^2+....C(k,k−1)a^1b^(k−1)+C(k,k)a^0b^k.`

Note that, for example, C(4,0)=C(4,4)=1,C(4,1)=C(4,3)=4,C(4,2)=6
and Pascal's triangle gives a diagram representing binomial coefficients, namely:

                                           1
                                       1       1
                                   1      2      1
                               1      3       3     1
                           1      4      6      4     1
                        1     5     10     10     5    1
                     1    6     15    20    15    6    1

etc.
The sequence of numbers in the i-th row are, respectively, C(1,0),C(i,1),C(i,2),C(i,3), etc.

The number C(i,j) is the sum of the two numbers C(i−1,j−1) and C(i−1,j) just above it in the diagram and to the left and right, respectively.

In this example, `a=2x`, `b= -y`, and k = 3.