  We study the 't Hooft expansion of $d=4$ $\mathcal{N}=4$ supersymmetric
Yang-Mills (SYM) theory with the gauge group $SO(N)$ or $Sp(N)$. We consider
the $1/N_5$ expansion with fixed $g_sN_5$, where $g_s$ denotes the string
coupling of bulk type IIB string theory on $AdS_5\times\mathbb{RP}^5$ and $N_5$
refers to the RR 5-form flux through $\mathbb{RP}^5$. $N_5$ differs from $N$
due to a shift coming from the RR charge of O3-plane. As an example, we
consider the $1/N_5$ expansion of the free energy of $\mathcal{N}=4$ SYM on
$S^4$ and the $1/2$ BPS circular Wilson loops in the fundamental representation
of $SO(N)$ or $Sp(N)$. We find that the $1/N_5$ expansion is more "closed
string like" than the ordinary $1/N$ expansion.
