Question

How do you use the binomial `(2v+3)^6` using Pascal's triangle?

Answer 1

`64v^6+576v^5+2160v^4+4320v^3+4860v^2+2916v+729`

Explanation:

Expand using the corresponding row of coefficients from Pascal's triangle for n = 6

`rArr1color(white)(xx)6color(white)(xx)15color(white)(xx)20color(white)(xx)15color(white)(xx)6color(white)(xx)1`

with `color(blue)"decreasing powers" " of "2v" from " (2v)^6to(2v)^0`

`"and "color(blue)" increasing powers"" of " 3" from " 3^0to3^6`

`1.(2v)^6. 3^0+6.(2v)^5. 3^1+15.(2v)^4. 3^2+20.(2v)^3. 3^3`

`+15.(2v)^2. 3^4+6.(2v)^1. 3^5+1.(2v)^0. 3^6`

`=64v^6+576v^5+2160v^4+4320v^3+4860v^2+2916v+729`