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15 year mortgage vs 30 year paid off in 15
Your calculations are correct if you use the same mortgage rate for both the 15 and 30 year mortgages.  However, generally when you apply for a 15 year mortgage the interest rate is significantly less than the 30 year rate.  The rate is lower for a number of reasons but mainly there is less risk for the bank on a 15 year payoff plan.