The machine that records the number of visitors to a museum marks 1879564. Note that this number has all distinct digits. What is the smallest number of additional visitors needed for the machine to record another number that also has all its digits distinct?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52
The correct option is (c).

Notice that the only digits that do not appear in the number 1879564 are 0, 2, and 3. The next number with all distinct digits will occur when the hundreds digit changes and we have 18796 _ _. Therefore, the smallest number will be 1879602, and there are still $1879602-1879564=38$ visitors remaining.