I know, $|S_4 |=24 =2^3\cdot3$.
So here the order of sylow $2$ subgroup is $8$ and by the third sylow theorem we can say the number of sylow $2$ subgroups is $1$ or $3$. Then by finding the sylow $2$ subgroups explicitly we can conclude that its $3$. But here we also know the number of sylow $3$ subgroups is $1$ or $4$.
My question is can we find the exact number of sylow $2$ and $3$ subgroups without calculating the the sylow subgroups explicitly, just by element counting of the subgroups, because sometimes we need only the exact number in of sylow subgroups ?
Suppose $S_4$ has a unique Sylow $2$-subgroup say $K$. By Sylow Second Theorem, $K$ must be normal in $S_4$. But $S_4$ does not have any normal subgroup of order $8$ (Refer here). Therefore the number of Sylow $2$-subgroups in $S_4$ must be three.