$(1+x+x^2+cdots+x^{100})^3$展开式中$x^{150}$前的系数为_____
解答:$(1+x+x^2+cdots+x^{100})^3=(1-x^{101})^3sumlimits_{k=0}^{+infty}C_{k+2}^2x^k$$=(1-3x^{101}+3x^{202}-x^{303})sumlimits_{k=0}^{+infty}C_{k+2}^2x^k$所以$x^{150}$前的系数为$C_{152}^2-3C_{51}^2$
