If a polynomial ideal can be generated by $k$ elements, can it be generated by $k$ elements of any generating set?

Question

Let $I = (p_1,\ldots, p_k) \subset \mathbb{C}[x_1,\ldots,x_n]$.

If we have a set of $k'$ polynomials such that $(q_1,\ldots,q_{k'}) = I$, can we always find a $k$-member subset such that $(q_{\sigma_1},\ldots,q_{\sigma_k}) = I$?

Answer 1

$(1) = (X-2,X-1) \ne (X-2)$ and $\ne(X-3)$.