The binomial coefficients satisfy a rich collection of identities, most derivable from (1+x)^n by substitution or calculus.

Substitution identities: x=1 gives sum = 2^n. x=-1 gives alternating sum = 0. Adding/subtracting: even-indexed sum = odd-indexed sum = 2^{n-1}. x=i gives real part = C_0 - C_2 + C_4 - ... = 2^{n/2}cos(npi/4). Cube roots of unity filter: (2^n + (1+omega)^n + (1+omega^2)^n)/3 = C_0 + C_3 + C_6 + ...

Differentiation identities: Differentiating (1+x)^n gives n(1+x)^{n-1} = sum rC(n,r)x^{r-1}. At x=1: n2^{n-1} = sum rC(n,r). Second derivative at x=1: n(n-1)*2^{n-2} = sum r(r-1)C(n,r). Combined: sum r^2C(n,r) = n(n+1)*2^{n-2}.

Integration identity: Integrating from 0 to 1 gives (2^{n+1}-1)/(n+1) = sum C(n,r)/(r+1). Integrating (1-x)^n gives 1/(n+1) = sum (-1)^r*C(n,r)/(r+1).

Product identities: Vandermonde's identity C(m+n,r) = sum C(m,k)*C(n,r-k). Special case: sum C(n,r)^2 = C(2n,n). Hockey stick: sum_{k=r}^{n} C(k,r) = C(n+1,r+1).

The key rC(n,r) = nC(n-1,r-1) identity converts weighted sums into standard sums, making it one of the most useful tools. Similarly, r(r-1)*C(n,r) = n(n-1)*C(n-2,r-2).