- #1
Jhenrique
- 685
- 4
There is a trinomial theorem relationed with the Pascal's tetrahedron, so that...
[tex](x+y+z)^5=[/tex]
[tex]
\\ +01x^5y^0z^0+05x^4y^1z^0+10x^3y^2z^0+10x^2y^3z^0+05x^1y^4z^0+01x^0y^5z^0
\\ +05x^4y^0z^1+20x^3y^1z^1+30x^2y^2z^1+20x^1y^3z^1+05x^0y^4z^1
\\ +10x^3y^0z^2+30x^2y^1z^2+30x^1y^2z^2+10x^0y^3z^2
\\ +10x^2y^0z^3+20x^1y^1z^3+10x^0y^2z^3
\\ +05x^1y^0z^4+05x^0y^1z^4
\\ +01x^0y^0z^5
[/tex]
Well, when it comes the binomial coefficients, they are easily determined so:
[tex]\binom{5}{0}...\binom{5}{1}...\binom{5}{2}...\binom{5}{3}...\binom{5}{4}...\binom{5}{5}[/tex]
But this sequence is for a linear development (binomial theorem). And when it comes to a bilinear development (trinomial theorem), how will be the coefficients? Certainly, there will a product between binomial numbers, but how will be such scheme?
[tex](x+y+z)^5=[/tex]
[tex]
\\ +01x^5y^0z^0+05x^4y^1z^0+10x^3y^2z^0+10x^2y^3z^0+05x^1y^4z^0+01x^0y^5z^0
\\ +05x^4y^0z^1+20x^3y^1z^1+30x^2y^2z^1+20x^1y^3z^1+05x^0y^4z^1
\\ +10x^3y^0z^2+30x^2y^1z^2+30x^1y^2z^2+10x^0y^3z^2
\\ +10x^2y^0z^3+20x^1y^1z^3+10x^0y^2z^3
\\ +05x^1y^0z^4+05x^0y^1z^4
\\ +01x^0y^0z^5
[/tex]
Well, when it comes the binomial coefficients, they are easily determined so:
[tex]\binom{5}{0}...\binom{5}{1}...\binom{5}{2}...\binom{5}{3}...\binom{5}{4}...\binom{5}{5}[/tex]
But this sequence is for a linear development (binomial theorem). And when it comes to a bilinear development (trinomial theorem), how will be the coefficients? Certainly, there will a product between binomial numbers, but how will be such scheme?
