Out of 10 contestants, six are to be selected for the final round of a competition - HSC - SSCE Mathematics Extension 1 - Question 8 - 2020 - Paper 1

After selecting the 6 contestants, we need to arrange 4 of them in positions 1st, 2nd, 3rd, and 4th. The number of arrangements (permutations) of 4 contestants is given by: $P(n, r) = \frac{n!}{(n - r)!}$ Therefore, we have: $P(6, 4) = \frac{6!}{(6 - 4)!} = \frac{6!}{2!}$ This gives the total arrangements of the top 4 contestants.

The total ways to select and arrange the contestants can be calculated as: $Total = C(10, 6) \times P(6, 4) = \frac{10!}{6!4!} \times \frac{6!}{2!} = \frac{10!}{4!2!}$ Thus, the final answer is the option C: $\frac{10!}{4!2!}$.