= Solution
The orbit of $(1,\ldots,k)$ consists precisely of the ordered $k$-tuples of distinct elements of $\{1,\ldots,n\}$:
$$
\operatorname{Orb}(x)
=\{(x_1,\ldots,x_k):x_i\ne x_j\text{ for }i\ne j\}.
$$
It has size
$$
n(n-1)\cdots(n-k+1)=\frac{n!}{(n-k)!}.
$$
The stabilizer consists of permutations fixing $1,\ldots,k$ pointwise, while freely permuting the remaining letters:
$$
\operatorname{Stab}(x)\cong S_{n-k},
\qquad
|\operatorname{Stab}(x)|=(n-k)!.
$$
Thus
$$
|\operatorname{Orb}(x)|\,|\operatorname{Stab}(x)|
=\frac{n!}{(n-k)!}(n-k)!=n!=|S_n|,
$$
verifying orbit-stabilizer.
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