2010 AMC 12B Problem 4

Since $31=4\cdot7+3,$ the first three days of the month each occur five times, and the other four days occur four times. Mondays and Wednesdays are equal in number exactly when both fall in the five-time group or both fall in the four-time group. If the first day is Monday, the five-time days are Mon, Tue, Wed (both appear five times). If the first day is Thursday or Friday, the five-time days miss both Monday and Wednesday (both appear four times). Every other starting day includes exactly one of Monday or Wednesday. So the first day can be Monday, Thursday, or Friday, giving $3$ possibilities. Thus, the correct answer is B.