Cool Summation Problem

Essentially, the problem is asking to take all subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ , in each subset take the product of all elements, and finally evaluate the sum all of these products.

Consider the product $\displaystyle \prod_{k=1}^{10} (1+k) = (1+1)(1+2)(1+3)(1+4)(1+5)(1+6)(1+7)(1+8)(1+9)(1+10)$ .

Note that when expanding this product (without adding the binomials), every term obtained is accounted for in the summation in the problem, except for choosing the first 1 for all 10 binomials when expanding.

Hence, we have a bijection between the terms in the expansion of $\displaystyle \prod_{k=1}^{10} (1+k)$ and the terms in the problem.

The answer is then simply $\displaystyle \left( \prod_{k=1}^{10} (1+k) \right) -1 = \left( \prod_{k=2}^{11} k \right) -1 = 11! - 1 = 39916799$ .