Question

a local chess tournament gives medals for first, second, and third place. there are five students from midland high, three students from leasburg high, and six students from cassville high competing in the tournament. which statements are true? check all that apply. order matters in this scenario. there are 2,184 ways to select a first - place, second - place, and third - place winner. the probability that all three winners are from midland high is 0.0275. the probability that all three winners are from leasburg high is 0.0046. the probability that all three winners are from cassville high is 0.0549

Explanation:

Step1: Determine if order matters

Since there are distinct first, second, and third - place medals, order matters.

Step2: Calculate the number of ways to select winners

The total number of students is \(5 + 3+6=14\). The number of ways to select 3 winners (where order matters) is \(P(14,3)=\frac{14!}{(14 - 3)!}=\frac{14!}{11!}=14\times13\times12=2184\)

Step3: Calculate probability for Midland High

The number of ways to select 3 students from Midland High (where order matters) is \(P(5,3)=\frac{5!}{(5 - 3)!}=\frac{5!}{2!}=5\times4\times3 = 60\). The probability is \(\frac{60}{2184}\approx0.0275\)

Step4: Calculate probability for Leasburg High

The number of ways to select 3 students from Leasburg High (where order matters) is \(P(3,3)=\frac{3!}{(3 - 3)!}=3! = 6\). The probability is \(\frac{6}{2184}\approx0.0027\) (not \(0.0046\))

Step5: Calculate probability for Cassville High

The number of ways to select 3 students from Cassville High (where order matters) is \(P(6,3)=\frac{6!}{(6 - 3)!}=\frac{6!}{3!}=6\times5\times4=120\). The probability is \(\frac{120}{2184}\approx0.0549\)

Answer:

Order matters in this scenario.
There are 2,184 ways to select a first - place, second - place, and third - place winner.
The probability that all three winners are from Midland High is 0.0275.
The probability that all three winners are from Cassville High is 0.0549.