Question

question 5
at a competition with 6 runners, 2 medals are awarded for first and second place.
each medal is different. how many ways are there to award the medals?
a. 15
b. 30
c. 720
d. 36

question 6
there are 10 students in a class. the teacher chooses 3 students to go to the library. the order
in which they are chosen does not matter. how many ways are there to choose the students?
a. 100
b. 120
c. 720
d. 90

question 7
you and your mom enter a drawing with 3 different prizes. the prizes are awarded at random.
if a total of 9 people entered the drawing, in how many ways can you win first prize and your
mom win second prize?
a. 7
b. 1
c. 9
d. 2

question 8
you and your mom enter a drawing with 3 different prizes. a total of 10 people entered the
drawing, and prizes are awarded randomly.
there are 720 ways to award the prizes.
what is the probability that you win first prize and your mom wins second prize?
a. $\frac{10}{720}$
b. $\frac{8}{720}$
c. $\frac{1}{10}$
d. $\frac{1}{720}$

Explanation:

Question 5

Step1: Identify permutation formula

We use permutation \(P(n,k)=\frac{n!}{(n-k)!}\) where \(n=6\), \(k=2\)

Step2: Calculate permutation value

\(P(6,2)=\frac{6!}{(6-2)!}=\frac{6\times5\times4!}{4!}=6\times5=30\)

Question 6

Step1: Identify combination formula

We use combination \(C(n,k)=\frac{n!}{k!(n-k)!}\) where \(n=10\), \(k=3\)

Step2: Calculate combination value

\(C(10,3)=\frac{10!}{3!(10-3)!}=\frac{10\times9\times8\times7!}{3\times2\times1\times7!}=120\)

Question 7

Step1: Fix first two prizes

You win first, mom wins second; only 1 way for this pair.

Step2: Count third prize options

7 remaining people for third prize, so total ways: \(1\times1\times7=7\)

Question 8

Step1: Count favorable outcomes

Only 1 favorable outcome (you 1st, mom 2nd, any of 8 left for 3rd: \(1\times1\times8=8\))

Step2: Calculate probability

Probability = \(\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{8}{720}\)

Answer:

Question 5: B. 30
Question 6: B. 120
Question 7: A. 7
Question 8: B. $\frac{8}{720}$