For r = 0, 1, ..., 10, let Ar, Br, and Cr denote, respectively, the coefficients of xr in the expansions of (1+x)10,(1+x)20 and (1+x)30. Then ∑r=110 ArB10Br−C10Ar is equal to

Ar, Br and Cr denote, respectively, the coefficient of xr in the expansions of (1 + x)10, (1 + x)20 and (1 + x)30.∴ Ar=10Cr,Br=20Cr,Cr=30Cr∴∑r=110 ArB10Br−C10Ar=B10∑r=110 ArBr−C10∑r=110 Ar2=B10∑r=110 10Cr20Cr−C10∑r=110  10Cr2=B10∑r=110 10Cr20C20−r−C10∑r=110  10Cr2=B10∑r=010 10Cr20C20−r−1−C10∑r=010  10Cr2−1=B10 30C20−1−C10 20C10−1                   ∵nC02+nC12+nC22+…+nCn2=2nCn=B1030C20−C1020C10+C10−B10= 20C1030C20−30C1020C10+C10−B10=C10−B10