At a gymnastics meet, twenty gymnasts compete for first, second, and third place. How many ways can first, second, and third place be assigned? Third place has been announced. In how many ways can the remaining two places be assigned? Third and second places have been announced. In how many ways can first place be assigned?

Part A:

Given that at a gymnastics meet, twenty gymnasts compete for first, second, and third place. The number of ways first, second, and third place can be assigned is given by

[tex] ^{20}P_3= \frac{20!}{(20-3)!} \\ \\ = \frac{20!}{17!} =20\times19\times18 \\ \\ =6,840\ ways[/tex]



Part B:

Given that at a gymnastics meet, twenty gymnasts compete for first, second, and third place. If the third place has been announced the number of ways that the remaining two places can be assigned is given by :

[tex] ^{19}P_2= \frac{19!}{(19-2)!} \\ \\ = \frac{19!}{17!} =19\times18 \\ \\ =342\ ways[/tex]



Part C:

Given that at a gymnastics meet, twenty gymnasts compete for first, second, and third place. If the third and second places have been announced the number of ways that the first place can be assigned is given by :

[tex] ^{18}P_1= \frac{18!}{(18-1)!} \\ \\ = \frac{18!}{17!} =18\ ways[/tex]