determining the number of arrangements of elements\nat a gymnastics meet, twenty gymnasts compete for first…

determining the number of arrangements of elements\nat a gymnastics meet, twenty gymnasts compete for first, second, and third place.\nhow many ways can first, second, and third place be assigned?\nthird place has been announced. in how many ways can the remaining two places be assigned?\nthird and second places have been announced. in how many ways can first place be assigned?

determining the number of arrangements of elements\nat a gymnastics meet, twenty gymnasts compete for first, second, and third place.\nhow many ways can first, second, and third place be assigned?\nthird place has been announced. in how many ways can the remaining two places be assigned?\nthird and second places have been announced. in how many ways can first place be assigned?

Answer

Answer:

  1. 6840
  2. 342
  3. 18

Explanation:

Step1: Use permutation formula

The formula for permutations is $P(n,r)=\frac{n!}{(n - r)!}$, where $n$ is the number of elements and $r$ is the number of elements chosen at a time. Here $n = 20$ and $r=3$. So $P(20,3)=\frac{20!}{(20 - 3)!}=\frac{20!}{17!}=20\times19\times18 = 6840$.

Step2: Adjust $n$ and $r$

Since third - place is announced, $n = 19$ and $r = 2$. Then $P(19,2)=\frac{19!}{(19 - 2)!}=\frac{19!}{17!}=19\times18=342$.

Step3: Adjust $n$ and $r$ again

Since second and third - place are announced, $n = 18$ and $r = 1$. Then $P(18,1)=\frac{18!}{(18 - 1)!}=\frac{18!}{17!}=18$.