Of the 50 candidates who wrote a departmental paper, 30 cleared it with an average score of 75 and the other 20 did not clear it, averaging 60. What is the average score of all 50 candidates?

Add the two totals: $30\times 75=2250$ and $20\times 60=1200$, so the whole group scored 3450 over 50 candidates, giving 3450/50 = 69. 67.5 is the plain midpoint of 75 and 60, which would be right only if the two groups were equal in size - they are not, so the average must lean towards the bigger group. 70 comes from reading 30 and 20 as 2 to 1 instead of 3 to 2. 66 weights the wrong way round, giving the 60s the larger count.