How many onto (or surjective) functions are there from an $n$-element ($n \ge 2$) set to a 2-element set?

Let the domain set be $A$ with $|A| = n$ and the codomain set be $B = \{b_1, b_2\}$ with $|B| = 2$.

1. The total number of functions from $A$ to $B$ is:

   $$|B|^{|A|} = 2^n$$

2. A function is not onto (not surjective) if its range does not cover the entire codomain $B$. In this case, that happens when all elements of $A$ map to only $b_1$, or all elements map to only $b_2$.

3. There are exactly $2$ such constant functions:

   • $f(x) = b_1$ for all $x \in A$

   • $f(x) = b_2$ for all $x \in A$

4. Therefore, the number of onto functions is:

   $$\text{Total functions} - \text{Non-onto functions} = 2^n - 2$$

Hence, (C) is the correct option.