Pascal's Rule: C(n,r) = C(n-1,r-1) + C(n-1,r). This recursive formula builds Pascal's triangle row by row.
Hockey stick identity: C(r,r) + C(r+1,r) + C(r+2,r) + ... + C(n,r) = C(n+1,r+1). This is the sum along a diagonal of Pascal's triangle.
Vandermonde's identity: C(m+n,r) = C(m,k)*C(n,r-k). Combinatorial proof: choosing r items from m+n items = choosing k from first m and r-k from last n.
Sum of squares: C(n,0)^2 + C(n,1)^2 + ... + C(n,n)^2 = C(2n,n). This follows from Vandermonde with m = n, r = n.