at a gymnastics meet, twenty gymnasts compete for first, second, and third place. how many ways can first…

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?

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?

Answer

Explanation:

Step1: Calculate initial number of ways

The number of ways to assign first, second and third - place out of 20 gymnasts is a permutation problem. The formula for permutations is $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 20$ and $r=3$. So $P(20,3)=\frac{20!}{(20 - 3)!}=\frac{20!}{17!}=20\times19\times18 = 6840$.

Step2: Calculate ways after third - place is announced

After third - place is announced, there are 19 gymnasts left for first and second place. Using the permutation formula with $n = 19$ and $r = 2$, $P(19,2)=\frac{19!}{(19 - 2)!}=\frac{19!}{17!}=19\times18=342$.

Step3: Calculate ways after second and third - place are announced

After second and third - place are announced, there are 18 gymnasts left for first place. So there are 18 ways to assign first place.

Answer:

6840 342 18