The value of sum_{k=0}^{10} C(10,k) * 2^k is _________.

Question

Accepted Answer

sum_{k=0}^{10} C(10,k) * 2^k = 3^10 = 59049. This sum looks like a mix of binomial coefficients and powers of 2. The Binomial Theorem tells us that (x + y)^n = sum_{k=0}^{n} C(n,k) * x^(n-k) * y^k. If we set n=10, x=1, and y=2, the right-hand side becomes exactly our sum: sum_{k=0}^{10} C(10,k) * 1^(10-k) * 2^k. The left-hand side is then (1+2)^10 = 3^10 = 59049.