Any Sylow p subgroup P of a group of lie type contained in a Borel subgroup B and if P is a sylow p subgroup in B then B is its normalizer in G Since all Borel subgroups are conjugate, we conclude that the number of sylow subgroups are exactly the number of Borel subgroups of G.
Now for PSL(n,q PSL(2,q ) , it is well known that sylow p- subgroups of it are elementary abelian groups. So P can be considered as $ s$ - dimensional vector space over $F_{p}$ where $q= p^{s}$. So the number of -$p^{k}$ -subgroups of P are exactly the number of k- dimensional vector spaces of P.
The number of Borel subgroups of PGL(n,q) can be computed using the structure of a Borel subgroup.It is also a well known fact that B is product of sylow Psubgroup of G and s-copies of cyclic group of order q-1.Hence the order m of B is exactly $(q-1)^{s}$ times the order of P. Since B is normalizer of P, you can conclude that the number of Sylow p- subgroup is PGL(n,q)/m . Now since PSL(n,q) is obtained dividing PGL(n,q) to a subgroup of it with cardinality n, now you can find the exact number
