Question

quiz 12.5.3: permutations and combinations
question 1
find the value of $6!$
a. 36
b. 120
c. 720
d. 30

question 2
find the value of $\frac{10!}{(10-2)!}$.
a. 720
b. 45
c. 90
d. 80

question 3
there are 6 people in a raffle drawing. two raffle winners each win gift cards. each gift card is the same. how many ways are there to choose the winners?
decide if the situation involves a permutation or a combination, and then find the number of ways to choose the winners.
a. permutation; number of ways = 15
b. combination; number of ways = 30
c. combination; number of ways = 15
d. permutation; number of ways = 30

question 4
at a competition with 5 runners, 5 medals are awarded for first place through fifth place. each medal is different. how many ways are there to award the medals?
decide if the situation involves a permutation or a combination, and then find the number of ways to award the medals.
a. permutation; number of ways = 1
b. combination; number of ways = 120
c. permutation; number of ways = 120
d. combination; number of ways = 1

Explanation:

Question 1

Step1: Define factorial formula

$n! = n\times(n-1)\times\cdots\times1$

Step2: Calculate 6!

$6! = 6\times5\times4\times3\times2\times1 = 720$

Question 2

Step1: Simplify denominator

$(10-2)! = 8!$

Step2: Cancel common terms

$\frac{10!}{8!} = \frac{10\times9\times8!}{8!} = 10\times9 = 90$

Question 3

Step1: Identify selection type

Gift cards are identical, so order does not matter: this is a combination.

Step2: Apply combination formula

$C(n,k)=\frac{n!}{k!(n-k)!}$, where $n=6, k=2$
$C(6,2)=\frac{6!}{2!(6-2)!}=\frac{6\times5\times4!}{2\times1\times4!}=15$

Question 4

Step1: Identify selection type

Medals are different, so order matters: this is a permutation.

Step2: Apply permutation formula

$P(n,k)=\frac{n!}{(n-k)!}$, where $n=5, k=5$
$P(5,5)=\frac{5!}{(5-5)!}=\frac{5!}{0!}=5\times4\times3\times2\times1=120$

Answer:

1. C. 720
2. C. 90
3. C. Combination; number of ways = 15
4. C. Permutation; number of ways = 120