How many ambiguous dates exist? How many ambiguous dates are there in a year? An ambiguous date is one like 8/3/2007 which could either mean the 8th of March or the 3rd of August.
Is it right to say that 1/1/2007 must mean the first of January so there are 11 ambiguous dates for each month (1/2/2007, 1/3/2007, ..., 1/12/2007)?
So the answer is $11\times12=132$?

The important question here, is when is a date ambiguous?
A date is ambiguous if the day (of the month) and the month (of the year) can be confused for one another; when the day can be a month as well. 
Since there are 12 months in a year, the day can only be confused with the month if the day is in $[1, 12]$, so there are twelve days each month where we don't know which is which.
That makes for a total of $12 \times 12 = 144$ days.
But of those twelve days per month, there will be one day where day and month are equal. So while we don't know which is which, it doesn't matter.
That makes for a total of $12 \times 1 = 12$ days that aren't ambiguous after all.
So your final answer is $12 \times 12 - 12 = 132$ days.
