This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of those considered in [A. A. Agrachev, G. Stefani, and P. Zezza, SIAM J. Control Optim., 41 (2002), pp. 991-1014], [L. Poggiolini, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), pp. 1-23], and [L. Poggiolini and G. Stefani, Systems Control Lett., 53 (2004), pp. 269-279]. We require both the strict bang-bang Legendre condition and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory.