The value of  nC1+n+1C2+n+2C3+⋯+n+m−1Cm is equal to

nC1+n+1C2+n+2C3+⋯+n+m−1Cm              =nCn−1+n+1Cn−1+n+2Cn−1+⋯+n+m−1Cn−1= Coefficient of xn−1 in (1+x)n+(1+x)n+1+(1+x)n+2+⋯+(1+x)n+m−1= Coefficient of xn−1 in (1+x)n(1+x)m−1(1+x)−1= Coefficient of xn−1 in (1+x)m+n−(1+x)nx= Coefficient of xn in (1+x)m+n−(1+x)n=m+nCn−1Similarly, we can prove mC1+m+1C2+m+2C3+⋯+m+n−1Cn=m+nCm−1