The coefficient of xr   [0≤r≤(n−1)] in the expansion of (x+3)n−1+(x+3)n−2(x+2)+(x+3)n−3(x+2)2+…+(x+2)n−1 is

We have,(x+3)n−1+(x+3)n−2(x+2)+(x+3)n−3(x+2)2+…+(x+2)n−1=(x+3)n−(x+2)n(x+3)−(x+2)=(x+3)n−(x+2)n∵xn−anx−a=xn−1+xn−2a1+xn−3a2+…+an−1Therefore, the coefficient of x r in the given expression = coefficient of xr in (x+3)n−(x+2)n=nCr3n−r−nCr2n−r=nCr3n−r−2n−r