nC0.nC0 - n+1C1.nC1 + n+2C2.nC2 - .... is coefficient of xn in
(nC0(x + 1)n - nC1(x + 1)n+1 + nC2(x + 1)n+2-.....)
Now, nC0 (x + 1)n - nC1(x + 1)n+1 + nC2(x + 1)2-.....
= (x + 1)n (nC0 - nC1(x + 1) + nC2(x + 1)2 -....)
= (x + 1)n (1 - (x + 1)n) (\(\because\) (1 - x)n = nC0- nC1x + nC2x2-....)
= (-x)n (x + 1)n
= (-1)n xn (x +1)n
Hence, Coefficient of xn in (nC0(x + 1)n - nC1(x + 1)n+1 + nC2(x+1)2) is (-1)n
\(\therefore\) nC0.nC0 - n+1C1.nC1 + n+2C2.nC2-....=(-1)n
