At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is

To cast a vote, the voter may choose any non-empty subset of the 10 candidates, with the restriction that the subset can contain at most 4 names because only 4 candidates are to be elected.

Therefore, the voter can select:
• exactly 1 candidate, or
• exactly 2 candidates, or
• exactly 3 candidates, or
• exactly 4 candidates.

The number of ways to choose $$r$$ candidates out of 10 is the combination $$\binom{10}{r}$$. Hence, the required total number of voting patterns is

$$\binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \binom{10}{4}$$

Compute each term:
$$\binom{10}{1} = 10$$
$$\binom{10}{2} = 45$$
$$\binom{10}{3} = 120$$
$$\binom{10}{4} = 210$$

Add the results:
$$10 + 45 + 120 + 210 = 385$$

Hence, the voter has 385 different ways to cast a valid vote.

Option C which is: 385