How much would it cost to try every possible burger combination?

Here's an alternative way to justify Thomas Andrews' calculation without appealing to averages:

Order all burgers in pairs containing one selection and its exact opposite. The total cost of each such pair is $$7.99 + 9.48 + 7\cdot 0.89 + 10\cdot 0.99 + 3\cdot 1.49 + 4\cdot 0.39 $$

To count the number of different pairs, identify each pair by its one-patty member. There are $2^{7+10+3+4}$ different one-patty burgers, so this is also the number of pairs.

The average cost of a burger is easy to compute - it will be $$\frac{(7.99+9.48) + 7\cdot 0.89 + 10\cdot 0.99 + 3\cdot 1.49 + 4\cdot 0.39}2$$

The number of burgers is $2\cdot 2^7\cdot 2^{10}\cdot 2^3 \cdot 2^4$.

Now multiply.

Assuming you are allowed to choose "none" for the non-burger options.