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

First, we need to select 6 contestants from a pool of 10. This can be calculated using the combination formula:

$C(n, r) = \frac{n!}{r!(n-r)!}$

With n = 10 and r = 6:

$C(10, 6) = \frac{10!}{6! \cdot 4!}$

Then, we can simplify:

$C(10, 6) = \frac{10 \cdot 9 \cdot 8 \cdot 7}{4!}$.

Next, we need to arrange the selected 6 contestants, focusing on how many ways we can position the top 4 (1st, 2nd, 3rd, 4th). This is given by the permutation formula:

$P(n, r) = \frac{n!}{(n-r)!}$

So for r = 4:

$P(6, 4) = \frac{6!}{(6-4)!} = 6! / 2!$.