For Y=β₀+β₁X₁+β₂X₂+⋯+βₖXₖ+e, what are degrees of freedom of |t| with n observations?

In the regression $Y=\beta_0+\beta_1X_1+\dots+\beta_kX_k+e$, there are k slope coefficients plus the intercept, so k + 1 parameters are estimated. Each estimated parameter uses up one degree of freedom, leaving $n-k-1$ degrees of freedom for the t statistic. Using $n-k$ forgets the intercept, which is the usual slip.