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, coefficient of xr in the given expression is equal to coefficient of xr in [(x + 3)n - (x + 2)n], which is given by nCr3n−r−nCr2n−r=nCr3n−r−2n−r