The permutation group $S_n$ can be constructed using different number of generators. Generally, one can generate $S_n$ by using the set $\{(12),(13),...(1n)\}$ or the set $\{(12),(12...n)\}$, which clearly have different cardinality.
My question is, is the number of rules that they satisfy the same regardless of the number of generators one uses?
The answer to the naive statement is NO.
Take the set of generators $S = \{\sigma_i | 1\leqslant i < n\}$, where $\sigma_i$ is the transposition $(i\, (i+1))$. Consider the following set of relations :
$$R := \{\sigma _{i}^{2}=1 |1\leqslant i<n\} \cup \{\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i} |j\neq i\pm 1\} \cup \{(\sigma_i \sigma_{i+1})^3=1 | 1\leqslant i < n\}$$ Then $\mathcal{S}_n = < S \,|\, R>$. Now take $x := \sigma_1 \sigma_3$. If your set of generators is $S' := S \cup \{x\}$, you'll need to have an extra relation where $x$ occurs, eg $x = \sigma_1 \sigma_3$.