Question

fill in the blank 15 points
(a) twelve skiers are competing in the final round of the olympic freestyle skiing aerial competition. in how many ways can 3 of the skiers finish first, second, and third to win the gold, silver, and bronze medals? type your answer..
(b) a relay race team has 4 runners who run different parts of the race. there are 16 students on your track team. how many different ways can your coach select students to compete in the race? type your answer..
(c) the school yearbook has an editor - in - chief and an assistant editor - in - chief. the staff of the yearbook has 15 students. in how many different ways can students be chosen for these 2 positions? type your answer..
just a reminder with how to do the \in how many ways can the letters in a word be rearranged to create different words\.
how many different ways can the letters messages be arranged to create eight - letter \words\?
m - 1; e - 2; s - 3; a - 1; g - 1
\\(\frac{8!}{2!3!}=3,360\\)

Explanation:

Part (a)

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 3 skiers out of 12 for the first, second, and third positions. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 12 \) and \( r=3 \).

Step2: Apply the permutation formula

$$
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$$

Step1: Identify the problem type

This is a permutation problem as the order of the 4 runners in the relay race matters. We use the permutation formula \( P(n, r)=\frac{n!}{(n - r)!} \) with \( n = 16 \) and \( r = 4 \).

Step2: Apply the permutation formula

$$
LATEXBLOCK0
$$

Step1: Identify the problem type

This is a permutation problem since the order (editor - in - chief and assistant editor - in - chief) matters. We use the permutation formula \( P(n, r)=\frac{n!}{(n - r)!} \) with \( n = 15 \) and \( r=2 \).

Step2: Apply the permutation formula

$$
LATEXBLOCK0
$$

Answer:

1320

Part (b)