Type 1 -- Direct substitution: sum C(n,r)*$a^r$ = (1+a)^n.

Type 2 -- Weighted sums: sum rC(n,r) = n2^{n-1}. Derived by differentiating (1+x)^n and putting x = 1.

Type 3 -- Products of binomial coefficients: sum C(n,r)*C(n,n-r) = C(2n,n). More generally, sum C(m,r)*C(n,k-r) = C(m+n,k) (Vandermonde).

Type 4 -- Alternating weighted sums: sum (-1)^r * r * C(n,r) = 0 (differentiate (1+x)^n, put x = -1).

Type 5 -- Sums with denominators: sum C$\frac{n,r}{(r+1)}$ = $\frac{2^{n+1}-1}{(n+1)}$ (integrate (1+x)^n from 0 to 1).

Type 6 -- Using partial fractions or telescoping: $\frac{1}{C(n,r}$) sums can sometimes be evaluated using beta function or telescoping identities.