Who Should We Choose?

The problem is basically asking the value of the following expression:

${18 \choose 1} + {18 \choose 2} + {18 \choose 3} + {18 \choose 4} + ... + {18 \choose 18}$

This can be calculated by adding the elements in the $18th$ row of Pascal's Triangle, and subtracting 1 from it (There can be no team of 0).

The sum of the elements in the $nth$ row of Pascal's Triangle can be expressed as $2^{n}$

In this case, $n = 18$

$2^{18} - 1$ = $\boxed {262143}$