Well I really don't get the question but I'll answer it according to my under standing
let say we got this binomial equation in terms of x and y
[tex](x+y)^n=0[/tex]
first the number of terms is the expansion is n+1
the first term is [tex]x^n[/tex] and the last term is [tex]y^n[/tex]
the exponent of [tex]x[/tex] descends linearly from n to 0
the exponent of [tex]y[/tex] descends linearly from 0 to n
the sum of the exponents of a and b in any of the terms is equal to n,
the coefficient of the second term and the second to the last term is n,
so
[tex](x+y)^n=(x^ny^0+n x^{n-1} y^1+ \frac{n}{2} x^{n-2} y^2+...+ \frac{n}{n-1}x^1 y^{n-1} + \frac{n}{n} x^{0} y^{n} )[/tex]
example:
[tex](x+y)^3[/tex]
in this example n = 3
the number of therms is 3 + 1 = 4
so there are 4 terms in this example
[tex] (x+y)^{3} = [/tex][tex]( x^{3}y^0 +3 x^{3-1} y^{1} +3x y^{3-1} + \frac{3}{3}y^3) [/tex]
[tex] (x+y)^{3} = x^{3} y^{0} +3 x^{2} y^{1} +3 x^{1} y^{2} + x^{0} y^{3} [/tex]
see the pattern in which the exponent of x is decreasing from 3 to 0
and the pattern in which the exponent of y is increasing from 0 to 3
this is the pattern that I knew in multiplying binomial algebraic expression
don't know if this is the answer you are looking for.
hit thank you if this helps you..
