The question is asking us to find the total number of times the digit 6 appears when we write down all the numbers from 501 to 700. To answer this we divide them into separate cases: 600-699 and 501-599 & 700.

In the first case: For 600-699, in 100 numbers, 6 can appear in tens place in 10 numbers (i.e., 60X, X being any digit from 0 to 9) and units place in another 10 numbers (i.e., 6XX, XX being any two digits from 00 to 99). In total, the digit 6 will be used 20 times per every hundred numbers from 600-699, meaning 120 occurrences in that range.

In the second case: For 501-599 and 700, 6 can appear only in units place 20 times (i.e., 50X6 and 70X6, where X can be any digit from 0 to 9).

This means the digit 6 will appear a total of 120 + 20 = 140 times. Therefore, Option 3 is correct.