Pascal's Identity: C(n,r) + C(n,r-1) = C(n+1,r). Vandermonde's Identity: C(m+n,r) = sum C(m,k)*C(n,r-k) for k=0 to r. Sum of row: C(n,0) + C(n,1) + ... + C(n,n) = 2^n. Alternating sum: C(n,0) - C(n,1) + ... = 0. These are useful for simplifying complicated counting expressions in JEE.