Question

Find the number of solutions of the equation $x+y+z=6.$ where $x,y,zϵW.$

Answer 1

Number of solutions=coefficient of $x^6$ in ${(1+x+x^2+...+x^6)}^3$
=coefficient of $x^6$ in ${(1-x^7)}^3{(1-x)}^{-3}$
=coefficient of $x^6$ in ${(1-x)}^{-3}$
${=}^{3+6-1}{\text{C}}_2{=}^8{\text{C}}_2=28.$