Question

question 23
there are 6 seniors on student council. two of them will be chosen to go to an all-district meeting. how many ways are there to choose the students who will go to the meeting?
decide if this is a permutation or a combination, and then find the number of ways to choose students who go.
a. combination; number of ways = 15
b. combination; number of ways = 30
c. permutation; number of ways = 30
d. permutation; number of ways = 15

question 24
at a competition with 7 runners, medals are awarded for first, second, and third places. each the 3 medals is different. how many ways are there to award the medals?
decide if this is a permutation or a combination, and find the number of ways to award the medals.
a. permutation; number of ways = 35
b. combination; number of ways = 35
c. combination; number of ways = 210
d. permutation; number of ways = 210

question 25
there are 11 paintings at an art show. three of them are chosen randomly to display in the gallery window. the order in which they are chosen does not matter. how many ways are there to choose the paintings?
a. 165
b. 121
c. 33
d. 990

Explanation:

(Question 23):

Step1: Identify combination (order no matter)

We use combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$, where $n=6$, $k=2$.

Step2: Calculate factorials and simplify

$$\begin{align*} C(6,2)&=\frac{6!}{2!(6-2)!}\\ &=\frac{6\times5\times4!}{2\times1\times4!}\\ &=\frac{30}{2}\\ &=15 \end{align*}$$

(Question 24):

Step1: Identify permutation (order matters)

We use permutation formula $P(n,k)=\frac{n!}{(n-k)!}$, where $n=7$, $k=3$.

Step2: Calculate factorials and simplify

$$\begin{align*} P(7,3)&=\frac{7!}{(7-3)!}\\ &=\frac{7\times6\times5\times4!}{4!}\\ &=7\times6\times5\\ &=210 \end{align*}$$

(Question 25):

Step1: Identify combination (order no matter)

We use combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$, where $n=11$, $k=3$.

Step2: Calculate factorials and simplify

$$\begin{align*} C(11,3)&=\frac{11!}{3!(11-3)!}\\ &=\frac{11\times10\times9\times8!}{3\times2\times1\times8!}\\ &=\frac{990}{6}\\ &=165 \end{align*}$$

Answer:

Question 23: A. Combination; number of ways = 15
Question 24: D. Permutation; number of ways = 210
Question 25: A. 165