How to expand 3 variables using Pascal's triangle?
#(x+y+z)^3#
1 Answer
It does not help much, but...
Explanation:
The instant response to this question might be to say that Pascal's triangle does not help, since it is concerned with powers of binomials. For example, the row
#(x+y)^3 = x^3+3x^2y+3xy^2+y^3#
What we can do is to combine the results of applying Pascal's triangle as follows:
If
#(x+y+z)^3 = (y+z)^3 = y^3+3y^2z+3yz^2+z^3#
If
#(x+y+z)^3 = (z+x)^3 = z^3+3z^2x+3zx^2+x^3#
If
#(x+y+z)^3 = (x+y)^3 = x^3+3x^2y+3xy^2+y^3#
The three expressions on the right hand side are essentially the full expression we are looking for, with some terms missing. The first one is missing any terms involving
#(x+y+z)^3 = x^3+y^3+z^3+3x^2y+3xy^2+3y^2z+3yz^2+3z^2x+3zx^2+kxyz#
for some
Note that the sum of the coefficients must be
#k = 27-(1+1+1+3+3+3+3+3+3) = 6#