In a dog show, there will be a gold and silver medal awarded.There are 6 finalists. In how many ways can the 2 medals be awarded

Answer:30Step-by-step explanation:Given: There are six finalists and two medals. That is it asks for the possibilities of 6 finalists taking two at a time. Say, for example the first and third finalist or the second and the sixth finalist and so on. The number of possibilities is given by [tex]$ ^nP_r $[/tex]                        [tex]$ ^nP_r = \frac{n!}{(n - r)!} $[/tex]So the number of possibilities of 6 finalists taken two at a time is [tex]$ ^6P_2 $[/tex][tex]$ \implies \frac{6!}{(6 - 2)!} = \frac{6!}{4!}  \hspace{15mm} [n! = n \times (n-1) \times (n - 2) \times . . . 2 \times 1] $[/tex][tex]$ \implies \frac{6 \times 5 \times 4 \times \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1}  = 6 \times 5 = 30 $[/tex]