The following is from exercise 1.5.14 of Permutation Group by Dixon and Mortimer and I'm not sure how to answer it.
Let $G \leq S_{n} $. Show that if $G$ has $r$ orbits when acting on $ \{ 1 , \dots , n \} $, then it is generated by at most $n - r $ elements.
Also, I noticed that there is a partial answer to the same question on the following link. A problem of permutation group
At the end of the answer however it mentions that by looking at a stabiliser $G_{ \alpha} $ and then looking at the orbits of that stabiliser you can see that $G$ can be generated from the generators of $G _{ \alpha } $ as well as some other elements $G$ to give the required number of generators for $G$. It says that the other elements of $G$ that generate the group can be found by looking carefully at the orbits of $G$ and smaller orbits of $G_{ \alpha } $.
I am not sure how the orbits of $G _{\alpha } $ and $G$ show that we can find the correct number of generators.